Theorems in algebraic number theoryFrom Wikipedia, the free encyclopedia

Contents

1 AlbertBrauerHasseNoether theorem 11.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 AnkenyArtinChowla congruence 32.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Bauerian extension 43.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 BrauerSiegel theorem 54.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5 Chebotarevs density theorem 65.1 History and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.2 Relation with Dirichlets theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.4 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4.1 Eective Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.4.2 Innite extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.5 Important consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6 Chebotaryov theorem on roots of unity 106.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7 Dirichlets unit theorem 12

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7.1 The regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7.2 Higher regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 Stark regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.4 p-adic regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 FerreroWashington theorem 168.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.2 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

9 Gras conjecture 189.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

10 GrossKoblitz formula 1910.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1910.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

11 GrunwaldWang theorem 2011.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2011.2 Wangs counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

11.2.1 An element that is an nth power almost everywhere locally but not everywhere locally . . . . 2011.2.2 An element that is an nth power everywhere locally but not globally . . . . . . . . . . . . . 21

11.3 A consequence of Wangs counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.4 Special elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.5 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.6 Explanation of Wangs counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2211.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2211.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2211.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

12 Hasse norm theorem 2312.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

13 Hasses theorem on elliptic curves 2413.1 Hasse-Weil Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2413.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2513.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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14 HasseArf theorem 2614.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

14.1.1 Higher ramication groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.1.2 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

14.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2714.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

15 HerbrandRibet theorem 2815.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2815.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

16 HermiteMinkowski theorem 3016.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

17 Hilberts Theorem 90 3117.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3117.2 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3117.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

18 HilbertSpeiser theorem 3318.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

19 KroneckerWeber theorem 3419.1 Field-theoretic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3419.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3419.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3419.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

20 Laorgues theorem 3620.1 Langlands conjectures for GL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3620.2 Representations of the Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3620.3 Automorphic representations of GLn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3620.4 Drinfelds theorem for GL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3620.5 Laorgues theorem for GLn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3620.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3720.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3720.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3720.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

21 Landau prime ideal theorem 3921.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3921.2 General number elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3921.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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21.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

22 Local Tate duality 4122.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

22.1.1 Case of nite modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4122.1.2 Case of p-adic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

22.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4222.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4222.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

23 Main conjecture of Iwasawa theory 4323.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4323.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4323.3 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4323.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

24 Mazurs control theorem 4524.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

25 Minkowskis bound 4625.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4625.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4625.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4625.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4625.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

26 MordellWeil theorem 4826.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4826.2 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4826.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4826.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

27 NeukirchUchida theorem 5027.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

28 Octic reciprocity 5128.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

29 Ostrowskis theorem 5229.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5229.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

29.2.1 Case I: n N |n| > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5329.2.2 Case II: n N |n| 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

29.3 Another Ostrowskis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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29.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5429.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

30 Principal ideal theorem 5630.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5630.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5630.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

31 Reection theorem 5831.1 Leopoldts Spiegelungssatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5831.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5931.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

32 Scholzs reciprocity law 6032.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6032.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

33 ShafarevichWeil theorem 6133.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6133.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

34 Shintanis unit theorem 6234.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6234.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

35 StarkHeegner theorem 6335.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6335.2 Real case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6435.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6435.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

36 Stickelbergers theorem 6536.1 The Stickelberger element and the Stickelberger ideal . . . . . . . . . . . . . . . . . . . . . . . . . 65

36.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6636.2 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6636.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6636.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6636.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6736.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

37 Takagi existence theorem 6837.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6837.2 A well-dened correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6837.3 Earlier work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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37.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6937.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6937.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

38 Yamamotos reciprocity law 7038.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7038.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 71

38.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7138.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7238.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 1

AlbertBrauerHasseNoether theorem

In algebraic number theory, theAlbertBrauerHasseNoether theorem states that a central simple algebra over analgebraic number eld K which splits over every completion Kv is a matrix algebra over K. The theorem is an exampleof a local-global principle in algebraic number theory and leads to a complete description of nite-dimensional divisionalgebras over algebraic number elds in terms of their local invariants. It was proved independently by Helmut Hasse,Richard Brauer, and Emmy Noether and by Abraham Adrian Albert.

1.1 Statement of the theoremLet A be a central simple algebra of rank d over an algebraic number eld K. Suppose that for any valuation v, Asplits over the corresponding local eld Kv:

AK Kv 'Md(Kv):

Then A is isomorphic to the matrix algebra Md(K).

1.2 ApplicationsUsing the theory of Brauer group, one shows that two central simple algebras A and B over an algebraic number eldK are isomorphic over K if and only if their completions Av and Bv are isomorphic over the completion Kv for everyv.Together with the GrunwaldWang theorem, the AlbertBrauerHasseNoether theorem implies that every centralsimple algebra over an algebraic number eld is cyclic, i.e. can be obtained by an explicit construction from a cycliceld extension L/K .

1.3 See also Class eld theory

Hasse norm theorem

1.4 References Albert, A.A.; Hasse, H. (1932), A determination of all normal division algebras over an algebraic number

eld, Trans. Amer. Math. Soc. 34 (3): 722726, doi:10.1090/s0002-9947-1932-1501659-x, Zbl 0005.05003

1

2 CHAPTER 1. ALBERTBRAUERHASSENOETHER THEOREM

Hasse, H.; Brauer, R.; Noether, E. (1931), Beweis eines Hauptsatzes in der Theorie der Algebren, Journalfr Mathematik 167: 399404

Fenster, D.D.; Schwermer, J. (2005), Delicate collaboration: Adrian Albert and Helmut Hasse and the Princi-pal Theorem in Division Algebras (PDF),Archive for history of exact sciences 59 (4): 349379, doi:10.1007/s00407-004-0093-6, retrieved 2009-07-05

Pierce, Richard (1982), Associative algebras, Graduate Texts in Mathematics 88, New York-Berlin: Springer-Verlag, ISBN 0-387-90693-2, Zbl 0497.16001

Reiner, I. (2003), Maximal Orders, London Mathematical Society Monographs. New Series 28, Oxford Uni-versity Press, p. 276, ISBN 0-19-852673-3, Zbl 1024.16008

Roquette, Peter (2005), The BrauerHasseNoether theorem in historical perspective (PDF), Schriften derMathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie derWissenschaften 15, MR 2222818,Zbl 1060.01009, CiteSeerX: 10 .1 .1 .72 .4101, retrieved 2009-07-05 Revised version Roquette, Peter (2013),Contributions to the history of number theory in the 20th century, Heritage of European Mathematics, Zrich:European Mathematical Society, pp. 176, ISBN 978-3-03719-113-2, Zbl 1276.11001

Albert, Nancy E. (2005), A Cubed & His Algebra, iUniverse, isbn-13: 978-0-595-32817-8

1.5 Notes

Chapter 2

AnkenyArtinChowla congruence

In number theory, theAnkenyArtinChowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artinand S. Chowla. It concerns the class number h of a real quadratic eld of discriminant d > 0. If the fundamental unitof the eld is

" =t+ u

pd

2

with integers t and u, it expresses in another form

ht

u(mod p)

for any prime number p > 2 that divides d. In case p > 3 it states that

2mhtu

X

0

Chapter 3

Bauerian extension

Bauers theorem redirects here. For the theorem in perturbation theory, see BauerFike theorem.

In mathematics, in the eld of algebraic number theory, a Bauerian extension is a eld extension of an algebraicnumber eld which is characterized by the prime ideals with inertial degree one in the extension.For a nite degree extension L/K of an algebraic number eld K we dene P(L/K) to be the set of primes p of Kwhich have a factor P with inertial degree one (that is, the residue eld of P has the same order as the residue eldof p).Bauers theorem states that if M/K is a nite degree Galois extension, then P(M/K) P(L/K) if and only if M L.In particular, nite degree Galois extensions N of K are characterised by set of prime ideals which split completelyin N.An extension F/K is Bauerian if it obeys Bauers theorem: that is, for every nite extension L of K, we have P(F/K) P(L/K) if and only if L contains a subeld K-isomorphic to F.All eld extensions of degree at most 4 over Q are Bauerian.[1] An example of a non-Bauerian extension is the Galoisextension of Q by the roots of 2x5 32x + 1, which has Galois group S5.[2]

3.1 See also Splitting of prime ideals in Galois extensions

3.2 References[1] Narkiewicz (1990) p.416

[2] Narkiewicz (1990) p.394

Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 86. ISBN 3-540-63003-1. Zbl 0819.11044.

Narkiewicz, Wadysaw (1990). Elementary and analytic theory of numbers (Second, substantially revised andextended ed.). Springer-Verlag. ISBN 3-540-51250-0. Zbl 0717.11045.

4

Chapter 4

BrauerSiegel theorem

In mathematics, the BrauerSiegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptoticresult on the behaviour of algebraic number elds, obtained by Richard Brauer and Carl Ludwig Siegel. It attemptsto generalise the results known on the class numbers of imaginary quadratic elds, to a more general sequence ofnumber elds

K1;K2; : : : :

In all cases other than the rational eld Q and imaginary quadratic elds, the regulator Ri of Ki must be taken intoaccount, because K then has units of innite order by Dirichlets unit theorem. The quantitative hypothesis of thestandard BrauerSiegel theorem is that if Di is the discriminant of Ki, then

[Ki : Q]

log jDij ! 0 as i!1:

Assuming that, and the algebraic hypothesis that Ki is a Galois extension of Q, the conclusion is that

log(hiRi)logpjDij ! 1 as i!1

where hi is the class number of Ki.This result is ineective, as indeed was the result on quadratic elds on which it built. Eective results in the samedirection were initiated in work of Harold Stark from the early 1970s.

4.1 References Richard Brauer, On the Zeta-Function of Algebraic Number Fields, American Journal ofMathematics 69 (1947),

243250.

5

Chapter 5

Chebotarevs density theorem

Chebotarevs density theorem in algebraic number theory describes statistically the splitting of primes in a givenGalois extension K of the eld Q of rational numbers. Generally speaking, a prime integer will factor into severalideal primes in the ring of algebraic integers of K. There are only nitely many patterns of splitting that may occur.Although the full description of the splitting of every prime p in a general Galois extension is a major unsolvedproblem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primesp less than a large integer N, tends to a certain limit as N goes to innity. It was proved by Nikolai Chebotaryov inhis thesis in 1922, published in (Tschebotare 1926).A special case that is easier to state says that if K is an algebraic number eld which is a Galois extension of Q ofdegree n, then the prime numbers that completely split in K have density

1/n

among all primes. More generally, splitting behavior can be specied by assigning to (almost) every prime numberan invariant, its Frobenius element, which strictly is a representative of a well-dened conjugacy class in the Galoisgroup

Gal(K/Q).

Then the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conju-gacy class with k elements occurs with frequency asymptotic to

k/n.

5.1 History and motivationWhen Carl Friedrich Gauss rst introduced the notion of complex integers Z[i], he observed that the ordinary primenumbers may factor further in this new set of integers. In fact, if a prime p is congruent to 1 mod 4, then it factorsinto a product of two distinct prime gaussian integers, or splits completely"; if p is congruent to 3 mod 4, then itremains prime, or is inert"; and if p is 2 then it becomes a product of the square of the prime (1+i) and the invertiblegaussian integer -i; we say that 2 ramies. For instance,

5 = (1 + 2i)(1 2i)

3

2 = i(1 + i)2

From this description, it appears that as one considers larger and larger primes, the frequency of a prime splittingcompletely approaches 1/2, and likewise for the primes that remain primes in Z[i]. Dirichlets theorem on arithmetic

6

5.2. RELATION WITH DIRICHLETS THEOREM 7

progressions demonstrates that this is indeed the case. Even though the prime numbers themselves appear rathererratically, splitting of the primes in the extension

Z Z[i]

follows a simple statistical law.Similar statistical laws also hold for splitting of primes in the cyclotomic extensions, obtained from the eld of rationalnumbers by adjoining a primitive root of unity of a given order. For example, the ordinary integer primes group intofour classes, each with probability 1/4, according to their pattern of splitting in the ring of integers corresponding tothe 8th roots of unity. In this case, the eld extension has degree 4 and is abelian, with the Galois group isomorphicto the Klein four-group. It turned out that the Galois group of the extension plays a key role in the pattern of splittingof primes. Georg Frobenius established the framework for investigating this pattern and proved a special case of thetheorem. The general statement was proved by Nikolai Grigoryevich Chebotaryov in 1922.

5.2 Relation with Dirichlets theoremThe Chebotarev density theorem may be viewed as a generalisation of Dirichlets theorem on arithmetic progressions.A quantitative form of Dirichlets theorem states that if N2 is an integer and a is coprime to N, then the proportionof the primes p congruent to a mod N is asymptotic to 1/n, where n=(N) is the Euler totient function. This is aspecial case of the Chebotarev density theorem for the Nth cyclotomic eld K. Indeed, the Galois group of K/Q isabelian and can be canonically identied with the group of invertible residue classes mod N. The splitting invariantof a prime p not dividing N is simply its residue class because the number of distinct primes into which p splitsis (N)/m, where m is multiplicative order of p modulo N; hence by the Chebotarev density theorem, primes areasymptotically uniformly distributed among dierent residue classes coprime to N.

5.3 FormulationLenstra & Stevenhagen (1996) give an earlier result of Frobenius in this area. Suppose K is a Galois extension of therational number eld Q, and P(t) a monic integer polynomial such that K is a splitting eld of P. It makes sense tofactorise P modulo a prime number p. Its 'splitting type' is the list of degrees of irreducible factors of P mod p, i.e. Pfactorizes in some fashion over the prime eld Fp. If n is the degree of P, then the splitting type is a partition of n.Considering also the Galois group G of K over Q, each g in G is a permutation of the roots of P in K; in other wordsby choosing an ordering of and its algebraic conjugates, G is faithfully represented as a subgroup of the symmetricgroup Sn. We can write g by means of its cycle representation, which gives a 'cycle type' c(g), again a partition of n.The theorem of Frobenius states that for any given choice of the primes p for which the splitting type of P mod pis has a natural density , with equal to the proportion of g in G that have cycle type .The statement of the more general Chebotarev theorem is in terms of the Frobenius element of a prime (ideal), whichis in fact an associated conjugacy class C of elements of the Galois group G. If we x C then the theorem says thatasymptotically a proportion |C|/|G| of primes have associated Frobenius element as C. When G is abelian the classesof course each have size 1. For the case of a non-abelian group of order 6 they have size 1, 2 and 3, and there arecorrespondingly (for example) 50% of primes p that have an order 2 element as their Frobenius. So these primes haveresidue degree 2, so they split into exactly three prime ideals in a degree 6 extension of Q with it as Galois group.[1]

5.4 StatementLet L be a nite Galois extension of a number eld K with Galois group G. Let X be a subset of G that is stable underconjugation. The set of primes v of K that are unramied in L and whose associated Frobenius conjugacy class F iscontained in X has density

#X#G :

[2]

8 CHAPTER 5. CHEBOTAREVS DENSITY THEOREM

5.4.1 Eective Version

The Generalized Riemann hypothesis implies an eective version[3] of the Chebotarev density theorem: if L/K isa nite Galois extension with Galois group G, and C a union of conjugacy classes of G, the number of unramiedprimes of K of norm below x with Frobenius conjugacy class in C is

jCjjGj

li(x) +Op

x(n logx+ log jj);where the constant implied in the big-O notation is absolute, n is the degree of L over Q, and its discriminant.

5.4.2 Innite extensions

The statement of the Chebotarev density theorem can be generalized to the case of an innite Galois extension L /K that is unramied outside a nite set S of primes of K (i.e. if there is a nite set S of primes of K such that anyprime of K not in S is unramied in the extension L / K). In this case, the Galois group G of L / K is a pronite groupequipped with the Krull topology. Since G is compact in this topology, there is a unique Haar measure on G. Forevery prime v of K not in S there is an associated Frobenius conjugacy class F. The Chebotarev density theorem inthis situation can be stated as follows:[2]

Let X be a subset of G that is stable under conjugation and whose boundary has Haar measure zero.Then, the set of primes v of K not in S such that F X has density

(X)

(G):

This reduces to the nite case when L / K is nite (the Haar measure is then just the counting measure).A consequence of this version of the theorem is that the Frobenius elements of the unramied primes of L are densein G.

5.5 Important consequencesThe Chebotarev density theorem reduces the problem of classifying Galois extensions of a number eld to that ofdescribing the splitting of primes in extensions. Specically, it implies that as a Galois extension of K, L is uniquelydetermined by the set of primes of K that split completely in it.[4] A related corollary is that if almost all prime idealsof K split completely in L, then in fact L = K.[5]

5.6 Notes

[1] This particular example already follows from the Frobenius result, because G is a symmetric group. In general, conjugacyin G is more demanding than having the same cycle type.

[2] Section I.2.2 of Serre

[3] Lagarias, J.C.; Odlyzko, A.M. (1977). Eective Versions of the Chebotarev Theorem. Algebraic Number Fields: 409464.

[4] Corollary VII.13.10 of Neukirch

[5] Corollary VII.13.7 of Neukirch

5.7. REFERENCES 9

5.7 References Lenstra, H. W.; Stevenhagen, P. (1996), Chebotarv and his density theorem, The Mathematical Intelligencer18: 2637, doi:10.1007/BF03027290, MR 1395088

Neukirch, Jrgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322,Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859

Serre, Jean-Pierre (1998) [1968], Abelian l-adic representations and elliptic curves (Revised reprint of the 1968original ed.), Wellesley, MA: A K Peters, Ltd., ISBN 1-56881-077-6, MR 1484415

Tschebotare, N. (1926), Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einergegebenen Substitutionsklasse gehren, Mathematische Annalen 95 (1): 191228, doi:10.1007/BF01206606

Chapter 6

Chebotaryov theorem on roots of unity

Not to be confused with Chebotarevs density theorem.

The theorem state that all submatrices of a DFT matrix of prime length are invertible.The Chebotaryov theorem on roots of unity was originally a conjecture made by Ostrowski in the context oflacunary series. Chebotaryov was the rst to prove it, in the 1930s. This proof involves tools from Galois theoryand did not please Ostrowski, who made comments arguing that it "does not meet the requirements of mathematicalesthetics" .[1]

Several proofs have been proposed since,[2] and it has even been discovered independently by Dieudonn.[3]

6.1 StatementLet be a matrix with entries aij = !ij ; 1 i; j n , where ! = e2i/n; n 2 N . If n is prime then any minor of

is non-zero.

6.2 ApplicationsFor signal processing purposes,[4] as a consequence of the Chebotaryov theorem on roots of unity, T. Tao statedan extension of the uncertainty principle.[5]

6.3 Notes[1] Stevenhagen et Al., 1996

[2] P.E. Frenkel,2003

[3] J. Dieudonn, 1970

[4] Cands, Romberg, Tao, 2006

[5] T. Tao, 2003

6.4 References Stevenhagen, Peter and Lenstra, Hendrik W (1996). Chebotarev and his density theorem. The Mathematical

Intelligencer (Springer) 18 (2): 2637. doi:10.1007/BF03027290. Frenkel, PE (2003). Simple proof of Chebotarevs theorem on roots of unity. ArXiv preprint math/0312398:

12398. arXiv:math/0312398. Bibcode:2003math.....12398F.

10

6.4. REFERENCES 11

Tao, Terence (2003). An uncertainty principle for cyclic groups of prime order. ArXiv preprint math/0308286:8286. arXiv:math/0308286. Bibcode:2003math......8286T.

Dieudonn,Jean (1970). Une proprit des racines de l'unit". Collection of articles dedicated to AlbertoGonzlez Domnguez on his sixty-fth birthday.

Candes, Emmanuel J and Romberg, Justin K and Tao, Terence; Romberg; Tao (2006). Stable signal recoveryfrom incomplete and inaccurate measurements. Communications on pure and applied mathematics (Wiley On-line Library) 59 (8): 12071223. arXiv:math/0503066. Bibcode:2005math......3066C. doi:10.1002/cpa.20124.

Chapter 7

Dirichlets unit theorem

In mathematics, Dirichlets unit theorem is a basic result in algebraic number theory due to Peter Gustav LejeuneDirichlet.[1] It determines the rank of the group of units in the ring OK of algebraic integers of a number eld K. Theregulator is a positive real number that determines how dense the units are.The statement is that the group of units is nitely generated and has rank (maximal number of multiplicatively inde-pendent elements) equal to

r = r1 + r2 1

where r1 is the number of real embeddings and r2 the number of conjugate pairs of complex embeddings of K. Thischaracterisation of r1 and r2 is based on the idea that there will be as many ways to embed K in the complex numbereld as the degree n = [K : Q]; these will either be into the real numbers, or pairs of embeddings related by complexconjugation, so that

n = r1 + 2r2.

Note that if K is Galois over Q then either r1 is non-zero or r2 is non-zero, but not both.Other ways of determining r1 and r2 are

use the primitive element theorem to write K = Q(), and then r1 is the number of conjugates of that arereal, 2r2 the number that are complex;

write the tensor product of elds K QR as a product of elds, there being r1 copies of R and r2 copies of C.

As an example, if K is a quadratic eld, the rank is 1 if it is a real quadratic eld, and 0 if an imaginary quadraticeld. The theory for real quadratic elds is essentially the theory of Pells equation.The rank is > 0 for all number elds besides Q and imaginary quadratic elds, which have rank 0. The 'size' of theunits is measured in general by a determinant called the regulator. In principle a basis for the units can be eectivelycomputed; in practice the calculations are quite involved when n is large.The torsion in the group of units is the set of all roots of unity of K, which form a nite cyclic group. For a numbereld with at least one real embedding the torsion must therefore be only {1,1}. There are number elds, for examplemost imaginary quadratic elds, having no real embeddings which also have {1,1} for the torsion of its unit group.Totally real elds are special with respect to units. If L/K is a nite extension of number elds with degree greaterthan 1 and the units groups for the integers of L and K have the same rank then K is totally real and L is a totallycomplex quadratic extension. The converse holds too. (An example is K equal to the rationals and L equal to animaginary quadratic eld; both have unit rank 0.)The theorem does not only applies to the maximal order OK but to any order O OK .[2]There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structureof the group of S-units, determining the rank of the unit group in localizations of rings of integers. Also, the Galoismodule structure of QOK;S Z Q has been determined.[3]

12

7.1. THE REGULATOR 13

7.1 The regulatorSuppose that u1,...,u are a set of generators for the unit group modulo roots of unity. If u is an algebraic number,write u1, ..., ur+1 for the dierent embeddings into R or C, and set Nj to 1, resp. 2 if corresponding embedding isreal, resp. complex. Then the r by r + 1 matrix whose entries are Nj log juji j has the property that the sum of anyrow is zero (because all units have norm 1, and the log of the norm is the sum of the entries of a row). This impliesthat the absolute value R of the determinant of the submatrix formed by deleting one column is independent of thecolumn. The number R is called the regulator of the algebraic number eld (it does not depend on the choice ofgenerators u). It measures the density of the units: if the regulator is small, this means that there are lots of units.The regulator has the following geometric interpretation. The map taking a unit u to the vector with entriesNj log juj jhas image in the r-dimensional subspace of Rr+1 consisting of all vector whose entries have sum 0, and by Dirichletsunit theorem the image is a lattice in this subspace. The volume of a fundamental domain of this lattice is R(r+1).The regulator of an algebraic number eld of degree greater than 2 is usually quite cumbersome to calculate, thoughthere are now computer algebra packages that can do it in many cases. It is usually much easier to calculate the producthR of the class number h and the regulator using the class number formula, and the main diculty in calculating theclass number of an algebraic number eld is usually the calculation of the regulator.

7.1.1 Examples

A fundamental domain in logarithmic space of the group of units of the cyclic cubic eld K obtained by adjoining to Q a root off(x) = x3 + x2 2x 1. If denotes a root of f(x), then a set of fundamental units is {1, 2} where 1 = 2 + 1 and 2 = 2 2. The area of the fundamental domain is approximately 0.910114, so the regulator of K is approximately 0.525455.

The regulator of an imaginary quadratic eld, or of the rational integers, is 1 (as the determinant of a 00matrix is 1).

The regulator of a real quadratic eld is the logarithm of its fundamental unit: for example, that of Q(5) islog((5 + 1)/2). This can be seen as follows. A fundamental unit is (5 + 1)/2, and its images under the twoembeddings into R are (5 + 1)/2 and (5 + 1)/2. So the r by r + 1 matrix is

"1 log

p5 + 1

2

; 1 logp5 + 1

2

#:

14 CHAPTER 7. DIRICHLETS UNIT THEOREM

The regulator of the cyclic cubic eld Q(), where is a root of x3 + x2 2x 1, is approximately 0.5255. Abasis of the group of units modulo roots of unity is {1, 2} where 1 = 2 + 1 and 2 = 2 2.[4]

7.2 Higher regulatorsA 'higher' regulator refers to a construction for a function on an algebraic K-group with index n > 1 that plays thesame role as the classical regulator does for the group of units, which is a group K1. A theory of such regulatorshas been in development, with work of Armand Borel and others. Such higher regulators play a role, for example,in the Beilinson conjectures, and are expected to occur in evaluations of certain L-functions at integer values of theargument.[5]

7.3 Stark regulatorThe formulation of Starks conjectures led Harold Stark to dene what is now called the Stark regulator, similar tothe classical regulator as a determinant of logarithms of units, attached to any Artin representation.[6][7]

7.4 p-adic regulatorLet K be a number eld and for each prime P of K above some xed rational prime p, let UP denote the local unitsat P and let U,P denote the subgroup of principal units in UP. Set

U1 =YP jp

U1;P :

Then let E1 denote the set of global units that map to U1 via the diagonal embedding of the global units in E.SinceE1 is a nite-index subgroup of the global units, it is an abelian group of rank r1+r21 . The p-adic regulatoris the determinant of the matrix formed by the p-adic logarithms of the generators of this group. Leopoldts conjecturestates that this determinant is non-zero.[8][9]

7.5 See also Elliptic unit Cyclotomic unit Shintanis unit theorem

7.6 Notes[1] Elstrodt 2007, 8.D

[2] PDF (Theorem 5.13)

[3] Neukirch, Schmidt & Wingberg 2000, proposition VIII.8.6.11.

[4] Cohen 1993, Table B.4

[5] Bloch, Spencer J. (2000). Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM MonographSeries 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. Zbl 0958.19001.

[6] PDF

[7] PDF

7.7. REFERENCES 15

[8] Neukirch et al (2008) p.626-627

[9] Iwasawa, Kenkichi (1972). Lectures on p-adic L-functions. Annals of Mathematics Studies 74. Princeton, NJ: PrincetonUniversity Press and University of Tokyo Press. pp. 3642. ISBN 0-691-08112-3. Zbl 0236.12001.

7.7 References Cohen, Henri (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics138. Berlin, New York: Springer-Verlag. ISBN 978-3-540-55640-4. MR 1228206. Zbl 0786.11071.

Elstrodt, Jrgen (2007). The Life and Work of Gustav Lejeune Dirichlet (18051859)" (PDF). Clay Mathe-matics Proceedings. Retrieved 2010-06-13.

Lang, Serge (1994). Algebraic number theory. Graduate Texts in Mathematics 110 (2nd ed.). New York:Springer-Verlag. ISBN 0-387-94225-4. Zbl 0811.11001.

Neukirch, Jrgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322,Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859

Neukirch, Jrgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, GrundlehrenderMathematischenWissenschaften 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, Zbl 0948.11001,MR 1737196

Chapter 8

FerreroWashington theorem

In algebraic number theory, the FerreroWashington theorem, proved rst by Ferrero & Washington (1979) andlater by Sinnott (1984), states that Iwasawas -invariant vanishes for cyclotomic Zp-extensions of abelian algebraicnumber elds.

8.1 HistoryIwasawa (1959) introduced the -invariant of a Zp-extension and observed that it was zero in all cases he calculated.Iwasawa & Sims (1966) used a computer to check that it vanishes for the cyclotomic Zp-extension of the rationals forall primes less than 4000. Iwasawa (1971) later conjectured that the -invariant vanishes for any Zp-extension, butshortly after Iwasawa (1973) discovered examples of non-cyclotomic extensions of number elds with non-vanishing-invariant showing that his original conjecture was wrong. He suggested, however, that the conjecture might stillhold for cyclotomic Zp-extensions.Iwasawa (1958) showed that the vanishing of the -invariant for cyclotomicZp-extensions of the rationals is equivalentto certain congruences between Bernoulli numbers, and Ferrero & Washington (1979) showed that the -invariantvanishes in these cases by proving that these congruences hold.

8.2 StatementFor a number eld K we let Km denote the extension by pm-power roots of unity, K^ the union of the Km and A(p)the maximal unramied abelian p-extension of K^ . Let the Tate module

Tp(K) = Gal(A(p)/K^) :

Then Tp(K) is a pro-p-group and so a Zp-module. Using class eld theory one can describe Tp(K) as isomorphic tothe inverse limit of the class groups Cm of the Km under norm.[1]

Iwasawa exhibited Tp(K) as a module over the completion Zp[[T]] and this implies a formula for the exponent of pin the order of the class groups Cm of the form

m+ pm + :

The FerreroWashington theorem states that is zero.[2]

8.3 References[1] Manin & Panchishkin 2007, p. 245

16

8.3. REFERENCES 17

[2] Manin & Panchishkin 2007, p. 246

Ferrero, Bruce; Washington, Lawrence C. (1979), The Iwasawa invariant vanishes for abelian numberelds, Annals of Mathematics. Second Series 109 (2): 377395, doi:10.2307/1971116, ISSN 0003-486X,MR 528968, Zbl 0443.12001

Iwasawa, Kenkichi (1958), On some invariants of cyclotomic elds, American Journal of Mathematics 81:280, doi:10.2307/2372857, ISSN 0002-9327, MR 0124317[http://www.jstor.org/stable/2372857 correction]

Iwasawa, Kenkichi (1959), On -extensions of algebraic number elds, Bulletin of the American Mathemat-ical Society 65 (4): 183226, doi:10.1090/S0002-9904-1959-10317-7, ISSN 0002-9904, MR 0124316

Iwasawa, Kenkichi (1971), On some innite Abelian extensions of algebraic number elds, Actes du CongrsInternational des Mathmaticiens (Nice, 1970), Tome 1, Gauthier-Villars, pp. 391394, MR 0422205

Iwasawa, Kenkichi (1973), On the -invariants of Z1-extensions, Number theory, algebraic geometry andcommutative algebra, in honor of Yasuo Akizuki, Tokyo: Kinokuniya, pp. 111, MR 0357371

Iwasawa, Kenkichi; Sims, Charles C. (1966), Computation of invariants in the theory of cyclotomic elds,Journal of the Mathematical Society of Japan 18: 8696, doi:10.4099/jmath.18.86, ISSN 0025-5645, MR0202700

Manin, Yu. I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathe-matical Sciences 49 (Second ed.), ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002

Sinnott, W. (1984), On the -invariant of the -transform of a rational function, Inventiones Mathematicae75 (2): 273282, doi:10.1007/BF01388565, ISSN 0020-9910, MR 732547, Zbl 0531.12004

Chapter 9

Gras conjecture

In algebraic number theory, the Gras conjecture (Gras 1977) relates the p-parts of the Galois eigenspaces of anideal class group to the group of global units modulo cyclotomic units. It was proved by Mazur & Wiles (1984) as acorollary of their work on the main conjecture of Iwasawa theory. Kolyvagin (1990) later gave a simpler proof usingEuler systems.

9.1 References Gras, Georges (1977), Classes d'idaux des corps abliens et nombres de Bernoulli gnraliss, Universit de

Grenoble. Annales de l'Institut Fourier 27 (1): 166, ISSN 0373-0956, MR 0450238

Kolyvagin, V. A. (1990), Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Boston,MA: Birkhuser Boston, pp. 435483, doi:10.1007/978-0-8176-4575-5_11, ISBN 978-0-8176-3428-5, MR1106906

Mazur, Barry; Wiles, Andrew (1984), Class elds of abelian extensions of Q", Inventiones Mathematicae 76(2): 179330, doi:10.1007/BF01388599, ISSN 0020-9910, MR 742853

18

Chapter 10

GrossKoblitz formula

In mathematics, the GrossKoblitz formula, introduced by Gross and Koblitz (1979) expresses a Gauss sum using aproduct of values of the p-adic gamma function. It is an analog of the ChowlaSelberg formula for the usual gammafunction. It implies the HasseDavenport relation and generalizes the Stickelberger theorem. Boyarsky (1980) gaveanother proof of the GrossKoblitz formula using Dworks work, and Robert (2001) gave an elementary proof.

10.1 StatementThe GrossKoblitz formula states that the Gauss sum can be given in terms of the p-adic gamma function p by

q(r) = sp(r)Y

0i

Chapter 11

GrunwaldWang theorem

In algebraic number theory, the GrunwaldWang theorem is a local-global principle stating thatexcept in someprecisely dened casesan element x in a number eld K is an nth power in K if it is an nth power in the completionKp for all but nitely many primes p of K. For example, a rational number is a square of a rational number if it isa square of a p-adic number for almost all primes p. The GrunwaldWang theorem is an example of a local-globalprinciple.It was introduced by Wilhelm Grunwald (1933), but there was a mistake in this original version that was found andcorrected by Shianghao Wang (1948). The theorem considered by Grunwald and Wang was more general than theone stated above as they discussed the existence of cyclic extensions with certain local properties, and the statementabout nth powers is a consequence of this.

11.1 HistorySome days later I was with Artin in his oce when Wang appeared. He said he had a counterexample to a lemmawhich had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself... Ofcourse he [Artin] was astonished, as were all of us students, that a famous theorem with two published proofs, one ofwhich we had all heard in the seminar without our noticing anything, could be wrong.John Tate, quoted in Roquette (2005, p.30)

Grunwald (1933), a student of Hasse, gave an incorrect proof of the erroneous statement that an element in a numbereld is an nth power if it is an nth power locally almost everywhere. Whaples (1942) gave another incorrect proof ofthis incorrect statement. However Wang (1948) discovered the following counter-example: 16 is a p-adic 8th powerfor all odd primes p, but is not a rational or 2-adic 8th power. In his doctoral thesis Wang (1950) written underArtin, Wang gave and proved the correct formulation of Grunwalds assertion, by describing the rare cases when itfails. This result is what is now known as the GrunwaldWang theorem. The history of Wangs counterexample isdiscussed in Roquette (2005, section 5.3)

11.2 Wangs counter-exampleGrunwalds original claim that an element that is an nth power almost everywhere locally is an nth power globally canfail in two distinct ways: the element can be an nth power almost everywhere locally but not everywhere locally, or itcan be an nth power everywhere locally but not globally.

11.2.1 An element that is an nth power almost everywhere locally but not everywherelocally

The element 16 in the rationals is an 8th power at all places except 2, but is not an 8th power in the 2-adic numbers.

20

11.3. A CONSEQUENCE OF WANGS COUNTEREXAMPLE 21

It is clear that 16 is not a 2-adic 8th power, and hence not a rational 8th power, since the 2-adic valuation of 16 is 4which is not divisible by 8.Generally, 16 is an 8th power in a eld K if and only if the polynomial X8 16 has a root in K. Write

X8 16 = (X4 4)(X4 + 4) = (X2 2)(X2 + 2)(X2 2X + 2)(X2 + 2X + 2):Thus, 16 is an 8th power in K if and only if 2, 2 or 1 is a square in K. Let p be any odd prime. It follows from themultiplicativity of the Legendre symbol that 2, 2 or 1 is a square modulo p. Hence, by Hensels lemma, 2, 2 or1 is a square in Qp .

11.2.2 An element that is an nth power everywhere locally but not globally16 is not an 8th power in Q(

p7) although it is an 8th power locally everywhere (i.e. in Qp(

p7) for all p). This

follows from the above and the equality Q2(p7) = Q2(

p1) .

11.3 A consequence of Wangs counterexampleWangs counterexample has the following interesting consequence showing that one cannot always nd a cyclic Galoisextension of a given degree of a number eld in which nitely many given prime places split in a specied way:There exists no cyclic degree 8 extensionK/Q in which the prime 2 is totally inert (i.e., such thatK2/Q2 is unramiedof degree 8).

11.4 Special eldsFor any s 2 let

s := exp2i

2s

+ exp

2i

2s

= 2 cos

2

2s

:

Note that the 2s th cyclotomic eld is

Q2s = Q(i; s):

A eld is called s-special if it contains s , but neither i , s+1 nor is+1 .

11.5 Statement of the theoremConsider a number eld K and a natural number n. Let S be a nite (possibly empty) set of primes of K and put

K(n; S) := fx 2 K j x 2 Knp for all p 62 Sg:The GrunwaldWang theorem says that

K(n; S) = Kn

unless we are in the special case which occurs when the following two conditions both hold:

1. K is s-special with an s such that 2s+1 divides n.

22 CHAPTER 11. GRUNWALDWANG THEOREM

2. S contains the special set S0 consisting of those (necessarily 2-adic) primes p such that Kp is s-special.

In the special case the failure of the Hasse principle is nite of order 2: the kernel of

K/Kn !Yp62S

Kp /Knp

is Z/2Z, generated by the element ns+1.

11.6 Explanation of Wangs counter-exampleThe eld of rational numbers K = Q is 2-special since it contains 2 = 0 , but neither i , 3 =

p2 nor i3 =

p2 .The special set is S0 = f2g . Thus, the special case in the GrunwaldWang theorem occurs when n is divisible by 8,and S contains 2. This explains Wangs counter-example and shows that it is minimal. It is also seen that an elementin Q is an nth power if it is a p-adic nth power for all p.The eld K = Q(

p7) is 2-special as well, but with S0 = ; . This explains the other counter-example above.[1]

11.7 See also The Hasse norm theorem states that for cyclic extensions an element is a norm if it is a norm everywhere locally.

11.8 Notes[1] See Chapter X of ArtinTate.

11.9 References Artin, Emil; Tate, John (1990), Class eld theory, ISBN 978-0-8218-4426-7, MR 0223335 Grunwald, W. (1933), Ein allgemeiner Existenzsatz fr algebraische Zahlkrper, Journal fr die reine und

angewandte Mathematik 169: 103107

Roquette, Peter (2005), The Brauer-Hasse-Noether theorem in historical perspective, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften [Publications of the Mathe-matics and Natural Sciences Section of Heidelberg Academy of Sciences] 15, Berlin, New York: Springer-Verlag, ISBN 978-3-540-23005-2

Wang, Shianghaw (1948), A counter-example to Grunwalds theorem, Annals of Mathematics. Second Series49: 10081009, ISSN 0003-486X, JSTOR 1969410, MR 0026992

Wang, Shianghaw (1950), On Grunwalds theorem, Annals of Mathematics. Second Series 51: 471484,ISSN 0003-486X, JSTOR 1969335, MR 0033801

Whaples, George (1942), Non-analytic class eld theory and Grnwalds theorem, Duke Mathematical Jour-nal 9 (3): 455473, doi:10.1215/s0012-7094-42-00935-9, ISSN 0012-7094, MR 0007010

Chapter 12

Hasse norm theorem

In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number elds, then if a nonzeroelement of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an elementk of K such that there is an element l of L with NL/K(l) = k ; in other words k is a relative norm of some elementof the extension eld L. To be a local norm means that for some prime p of K and some prime P of L lying over K,then k is a norm from LP; here the prime p can be an archimedean valuation, and the theorem is a statement aboutcompletions in all valuations, archimedean and non-archimedean.The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that3 is a local norm everywhere for the extensionQ(p3;p13)/Q but is not a global norm. Serre and Tate showed thatanother counterexample is given by the eldQ(

p13;

p17)/Qwhere every rational square is a local norm everywhere

but 52 is not a global norm.This is an example of a theorem stating a local-global principle.The full theorem is due to Hasse (1931). The special case when the degree n of the extension is 2 was proved byHilbert (1897), and the special case when n is prime was proved by Furtwangler (1902).The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2(L/K)is trivial if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the rst cohomology ofthe idele class group vanishes. This is true for all nite Galois extensions of number elds, not just cyclic ones. Forcyclic extensions the group H2(L/K) is isomorphic to the Tate cohomology group H0(L/K) which describes whichelements are norms, so for cyclic extensions it becomes Hasses theorem that an element is a norm if it is a local normeverywhere.

12.1 See also GrunwaldWang theorem, about when an element that is a power everywhere locally is a power.

12.2 References Hasse, H. (1931), Beweis eines Satzes und Wiederlegung einer Vermutung ber das allgemeine Normen-

restsymbol, Nachrichten von der Gesellschaft der Wissenschaften zu Gttingen, Mathematisch-PhysikalischeKlasse: 6469

H. Hasse, A history of class eld theory, in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory,Academic Press, 1973. Chap.XI.

G. Janusz, Algebraic number elds, Academic Press, 1973. Theorem V.4.5, p. 156

23

Chapter 13

Hasses theorem on elliptic curves

Hasses theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number ofpoints on an elliptic curve over a nite eld, bounding the value both above and below.If N is the number of points on the elliptic curve E over a nite eld with q elements, then Helmut Hasse's resultstates that

jN (q + 1)j 2pq:

That is, the interpretation is that N diers from q + 1, the number of points of the projective line over the same eld,by an 'error term' that is the sum of two complex numbers, each of absolute value q.This result had originally been conjectured by Emil Artin in his thesis.[1] It was proven by Hasse in 1933, with theproof published in a series of papers in 1936.[2]

Hasses theorem is equivalent to the determination of the absolute value of the roots of the local zeta-function of E.In this form it can be seen to be the analogue of the Riemann hypothesis for the function eld associated with theelliptic curve.

13.1 Hasse-Weil BoundA generalization of the Hasse bound to higher genus algebraic curves is the HasseWeil bound. This provides a boundon the number of points on a curve over a nite eld. If the number of points on the curve C of genus g over thenite eld Fq of order q is #C(Fq) , then

j#C(Fq) (q + 1)j 2gpq:

This result is again equivalent to the determination of the absolute value of the roots of the local zeta-function of C,and is the analogue of the Riemann hypothesis for the function eld associated with the curve.The HasseWeil bound reduces to the usual Hasse bound when applied to elliptic curves, which have genus g=1.The HasseWeil bound is a consequence of the Weil conjectures, originally proposed by Andr Weil in 1949.[3] Theproof was provided by Pierre Deligne in 1974.[4]

13.2 Notes[1] Artin, Emil (1924), Quadratische Krper im Gebiete der hheren Kongruenzen. II. Analytischer Teil, Mathematische

Zeitschrift 19 (1): 207246, doi:10.1007/BF01181075, ISSN 0025-5874, MR 1544652, Zbl 51.0144.05

[2] Hasse, Helmut (1936), Zur Theorie der abstrakten elliptischen Funktionenkrper. I, II & III, Crelles Journal 1936 (175),doi:10.1515/crll.1936.175.193, ISSN 0075-4102, Zbl 0014.14903

24

13.3. SEE ALSO 25

[3] Weil, Andr (1949), Numbers of solutions of equations in nite elds, Bulletin of the American Mathematical Society 55(5): 497508, doi:10.1090/S0002-9904-1949-09219-4, ISSN 0002-9904, MR 0029393

[4] Deligne, Pierre (1974), La Conjecture de Weil: I, PublicationsMathmatiques de l'IHS 43: 273307, doi:10.1007/BF02684373,ISSN 0073-8301, MR 340258, Zbl 0287.14001

13.3 See also SatoTate conjecture Schoofs algorithm

13.4 References Hurt, Norman E. (2003), Many Rational Points. Coding Theory and Algebraic Geometry, Mathematics and its

Applications 564, Dordrecht: Kluwer/Springer-Verlag, ISBN 1-4020-1766-9, MR 2042828,

Niederreiter, Harald; Xing, Chaoping (2009), Algebraic Geometry in Coding Theory and Cryptography, Prince-ton: Princeton University Press, ISBN 978-0-6911-0288-7, MR 2573098,

Chapter V of Silverman, Joseph H. (1994), The arithmetic of elliptic curves, Graduate Texts in Mathematics106, New York: Springer-Verlag, ISBN 978-0-387-96203-0, MR 1329092,

Washington, Lawrence C. (2008), Elliptic Curves. Number Theory and Cryptography, 2nd Ed, Discrete Math-ematics and its Applications, Boca Raton: Chapman & Hall/CRC Press, ISBN 978-1-4200-7146-7, MR2404461,

Chapter 14

HasseArf theorem

In mathematics, specically in local class eld theory, the HasseArf theorem is a result concerning jumps of theupper numbering ltration of the Galois group of a nite Galois extension. A special case of it was originally provedby Helmut Hasse,[1][2] and the general result was proved by Cahit Arf.[3][4]

14.1 Statement

14.1.1 Higher ramication groupsMain article: Ramication group

The theorem deals with the upper numbered higher ramication groups of a nite abelian extension L/K. So assumeL/K is a nite Galois extension, and that vK is a discrete normalised valuation of K, whose residue eld has charac-teristic p > 0, and which admits a unique extension to L, say w. Denote by vL the associated normalised valuationew of L and let O be the valuation ring of L under vL. Let L/K have Galois group G and dene the s-th ramicationgroup of L/K for any real s 1 by

Gs(L/K) = f 2 G : vL(a a) s+ 1 all for a 2 Og:So, for example, G is the Galois group G. To pass to the upper numbering one has to dene the function L/Kwhich in turn is the inverse of the function L/K dened by

L/K(s) =

Z s0

dx

jG0 : Gxj :

The upper numbering of the ramication groups is then dened by Gt(L/K) = Gs(L/K) where s = L/K(t).These higher ramication groups Gt(L/K) are dened for any real t 1, but since vL is a discrete valuation, thegroups will change in discrete jumps and not continuously. Thus we say that t is a jump of the ltration {Gt(L/K) : t 1} if Gt(L/K) Gu(L/K) for any u > t. The HasseArf theorem tells us the arithmetic nature of these jumps

14.1.2 Statement of the theoremWith the above set up, the theorem states that the jumps of the ltration {Gt(L/K) : t 1} are all rational integers.[4][5]

14.2 ExampleSuppose G is cyclic of order pn , p residue characteristic and G(i) be the subgroup of G of order pni . The theoremsays that there exist positive integers i0; i1; :::; in1 such that

26

14.3. NOTES 27

G0 = = Gi0 = G = G0 = = Gi0Gi0+1 = = Gi0+pi1 = G(1) = Gi0+1 = = Gi0+i1Gi0+pi1+1 = = Gi0+pi1+p2i2 = G(2) = Gi0+i1+1...Gi0+pi1++pn1in1+1 = 1 = G

i0++in1+1: [4]

14.3 Notes[1] H. Hasse, Fhrer, Diskriminante und Verzweigunsgskrper relativ Abelscher Zahlkrper, J. Reine Angew. Math. 162

(1930), pp.169184.

[2] H. Hasse, Normenresttheorie galoisscher Zahlkrper mit Anwendungen auf Fhrer und Diskriminante abelscher Zahlkrper,J. Fac. Sci. Tokyo 2 (1934), pp.477498.

[3] Arf, C. (1939). Untersuchungen ber reinverzweigte Erweiterungen diskret bewerteter perfekter Krper. J. Reine Angew.Math. (in German) 181: 144. Zbl 0021.20201.

[4] Serre (1979) IV.3, p.76

[5] Neukirch (1999) Theorem 8.9, p.68

14.4 References Neukirch, Jrgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322,

Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 Serre, Jean-Pierre (1979), Local elds, Graduate Texts in Mathematics 67, Translated from the French by

Marvin Jay Greenberg, Springer-Verlag, ISBN 0-387-90424-7, MR 554237, Zbl 0423.12016

Chapter 15

HerbrandRibet theorem

In mathematics, the HerbrandRibet theorem is a result on the class group of certain number elds. It is a strength-ening of Ernst Kummer's theorem to the eect that the prime p divides the class number of the cyclotomic eld ofp-th roots of unity if and only if p divides the numerator of the n-th Bernoulli number Bn for some n, 0 < n < p 1.The HerbrandRibet theorem species what, in particular, it means when p divides such an Bn.The Galois group of the cyclotomic eld of pth roots of unity for an odd prime p, Q() with p = 1, consists of thep 1 group elements a, where a() = a . As a consequence of the Fermats little theorem, in the ring of p-adicintegers Zp we have p 1 roots of unity, each of which is congruent mod p to some number in the range 1 to p 1; we can therefore dene a Dirichlet character (the Teichmller character) with values in Zp by requiring that forn relatively prime to p, (n) be congruent to n modulo p. The p part of the class group is a Zp -module (since it isp-primary), hence a module over the group ring Zp[] . We now dene idempotent elements of the group ring foreach n from 1 to p 1, as

n =1

p 1p1Xa=1

!(a)n1a :

It is easy to see that P n = 1 and ij = iji where ij is the Kronecker delta. This allows us to break up the ppart of the ideal class group G of Q() by means of the idempotents; if G is the ideal class group, then, letting Gn =n(G), we have G = Gn .The HerbrandRibet theorem states that for odd n, Gn is nontrivial if and only if p divides the Bernoulli numberBpn.[1] The part saying p divides Bpn if Gn is not trivial is due to Jacques Herbrand.[2] The converse, that if pdivides Bpn then Gn is not trivial is due to Kenneth Ribet, and is considerably more dicult. By class eld theory,this can only be true if there is an unramied extension of the eld of pth roots of unity by a cyclic extension ofdegree p which behaves in the specied way under the action of ; Ribet proves this by actually constructing such anextension using methods in the theory of modular forms. A more elementary proof of Ribets converse to Herbrandstheorem, a consequence of the theory of Euler systems, can be found in Washingtons book.[3]

The theorem makes no assertion about even values of n, but there is no known p for which Gn is nontrivial for anyeven n: triviality for all p would be a consequence of Vandivers conjecture.[4]

Ribets methods were pushed further by Barry Mazur and Andrew Wiles in order to prove the main conjecture ofIwasawa theory,[5] a corollary of which is a strengthening of the HerbrandRibet theorem: the power of p dividingBpn is exactly the power of p dividing the order of Gn.

15.1 See also

Iwasawa theory

28

15.2. NOTES 29

15.2 Notes[1] Ribet, Ken (1976). A modular construction of unramied p-extensions of Q ()". Inv. Math. 34 (3): 151162.

doi:10.1007/bf01403065.

[2] Herbrand, J. (1932). Sur les classes des corps circulaires. J. Math. Pures Appl., IX. Sr. (in French) 11: 417441. ISSN0021-7824. Zbl 0006.00802.

[3] Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields (Second ed.). New York: Springer-Verlag. ISBN0-387-94762-0.

[4] Coates, John; Sujatha, R. (2006). Cyclotomic Fields and Zeta Values. Springer Monographs in Mathematics. Springer-Verlag. pp. 34. ISBN 3-540-33068-2. Zbl 1100.11002.

[5] Mazur, Barry & Wiles, Andrew (1984). Class Fields of Abelian Extension of Q ". Inv. Math. 76 (2): 179330.doi:10.1007/bf01388599.

Chapter 16

HermiteMinkowski theorem

In mathematics, especially in algebraic number theory, the HermiteMinkowski theorem states that for any integerN there are only nitely many number elds, i.e., nite eld extensions K of the rational numbers Q, such that thediscriminant of K/Q is at most N. The theorem is named after Charles Hermite and Hermann Minkowski.This theorem is a consequence of the estimate for the discriminant

pjdK j n

n

n!

4

n/2where n is the degree of the eld extension, together with Stirlings formula for n!. This inequality also shows thatthe discriminant of any number eld strictly bigger than Q is not 1, which in turn implies that Q has no unramiedextensions.

16.1 ReferencesNeukirch, Jrgen (1999). Algebraic Number Theory. Springer. Section III.2

30

Chapter 17

Hilberts Theorem 90

In abstract algebra, Hilberts Theorem 90 (or Satz 90) is an important result on cyclic extensions of elds (or to oneof its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is a cyclic extensionof elds with Galois group G = Gal(L/K) generated by an element s and if a is an element of L of relative norm 1,then there exists b in L such that

a = s(b)/b.

The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's famous Zahlbericht (Hilbert1897, 1998), although it is originally due to Kummer (1855, p.213, 1861). Often a more general theorem due toEmmy Noether (1933) is given the name, stating that if L/K is a nite Galois extension of elds with Galois group G= Gal(L/K), then the rst cohomology group is trivial:

H1(G, L) = {1}

17.1 ExamplesLet L/K be the quadratic extension Q(i)/Q . The Galois group is cyclic of order 2, its generator s acting via conju-gation:

s : c di 7! c+ di :An element x = a + bi in L has norm xxs = a2 + b2 . An element of norm one corresponds to a rational solutionof the equation a2+ b2 = 1 or in other words, a point with rational coordinates on the unit circle. Hilberts Theorem90 then states that every element y of norm one can be parametrized (with integral c, d) as

y =c+ di

c di =c2 d2c2 + d2

+2cd

c2 + d2i

which may be viewed as a rational parametrization of the rational points on the unit circle. Rational points (x; y) =(a/c; b/c) on the unit circle x2+y2 = 1 correspond to Pythagorean triples, i.e. triples (a; b; c) of integers satisfyinga2 + b2 = c2 .

17.2 CohomologyThe theorem can be stated in terms of group cohomology: if L is the multiplicative group of any (not necessarilynite) Galois extension L of a eld K with corresponding Galois group G, then

H1(G, L) = {1}.

31

32 CHAPTER 17. HILBERTS THEOREM 90

A further generalization using non-abelian group cohomology states that if H is either the general or special lineargroup over L, then

H1(G,H) = {1}.

This is a generalization since L = GL1(L).Another generalization is H1et(X;Gm) = H1(X;OX) = Pic(X) for X a scheme, and another one to Milnor K-theory plays a role in Voevodskys proof of the Milnor conjecture.

17.3 References Hilbert, David (1897), Die Theorie der algebraischen Zahlkrper, Jahresbericht der DeutschenMathematiker-

Vereinigung (in German) 4: 175546, ISSN 0012-0456 Hilbert, David (1998), The theory of algebraic number elds, Berlin, New York: Springer-Verlag, ISBN 978-

3-540-62779-1, MR 1646901

Kummer, Ernst Eduard (1855), "ber eine besondere Art, aus complexen Einheiten gebildeter Ausdrcke.,Journal fr die reine und angewandte Mathematik (in German) 50: 212232, doi:10.1515/crll.1855.50.212,ISSN 0075-4102

Kummer, Ernst Eduard (1861), Zwei neue Beweise der allgemeinen Reciprocittsgesetze unter den Resten undNichtresten der Potenzen, deren Grad eine Primzahl ist, Abdruck aus den Abhandlungen der Kgl. Akademieder Wissenschaften zu Berlin (in German), Reprinted in volume 1 of his collected works, pages 699839

Chapter II of J.S. Milne, Class Field Theory, available at his website . Neukirch, Jrgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren

derMathematischenWissenschaften 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, Zbl 0948.11001,MR 1737196

Noether, Emmy (1933), Der Hauptgeschlechtssatz fr relativ-galoissche Zahlkrper., Mathematische An-nalen (in German) 108 (1): 411419, doi:10.1007/BF01452845, ISSN 0025-5831, Zbl 0007.29501

Snaith, Victor P. (1994), Galois module structure, Fields Institute monographs, Providence, RI: AmericanMathematical Society, ISBN 0-8218-0264-X, Zbl 0830.11042

Chapter 18

HilbertSpeiser theorem

In mathematics, the HilbertSpeiser theorem is a result on cyclotomic elds, characterising those with a normalintegral basis. More generally, it applies to any nite abelian extension ofQ, which by the KroneckerWeber theoremare isomorphic to subelds of cyclotomic elds.

HilbertSpeiser Theorem. A nite abelian extension K/Q has a normal integral basis if and only if itis tamely ramied over Q.

This is the condition that it should be a subeld of Q(n) where n is a squarefree odd number. This result wasintroduced by Hilbert (1897, Satz 132, 1998, theorem 132) in his Zahlbericht and by Speiser (1916, corollary toproposition 8.1).In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means ofGaussian periods. For example if we take n a prime number p > 2, Q(p) has a normal integral basis consisting of allthe p-th roots of unity other than 1. For a eld K contained in it, the eld trace can be used to construct such a basisin K also (see the article on Gaussian periods). Then in the case of n squarefree and odd, Q(n) is a compositumof subelds of this type for the primes p dividing n (this follows from a simple argument on ramication). Thisdecomposition can be used to treat any of its subelds.Cornelius Greither, Daniel R. Replogle, and Karl Rubin et al. (1999) proved a converse to the HilbertSpeisertheorem:

Each nite tamely ramied abelian extension K of a xed number eld J has a relative normal integralbasis if and only if J =Q.

18.1 References Greither, Cornelius; Replogle, Daniel R.; Rubin, Karl; Srivastav, Anupam (1999), Swan modules and Hilbert

Speiser number elds, doi:10.1006/jnth.1999.2425 Hilbert, David (1897), Die Theorie der algebraischen Zahlkrper, Jahresbericht der DeutschenMathematiker-

Vereinigung (in German) 4: 175546, ISSN 0012-0456 Hilbert, David (1998), The theory of algebraic number elds, Berlin, New York: Springer-Verlag, ISBN 978-

3-540-62779-1, MR 1646901 Speiser, A. (1916), Gruppendeterminante und Krperdiskriminante,Mathematische Annalen (Springer Berlin

/ Heidelberg) 77 (4): 546562, doi:10.1007/BF01456968, ISSN 0025-5831

33

Chapter 19

KroneckerWeber theorem

In algebraic number theory, it can be shown that every cyclotomic eld is an abelian extension of the rational numbereld Q. The KroneckerWeber theorem provides a partial converse: every abelian extension of Q is containedwithin some cyclotomic eld. In other words, every algebraic integer whose Galois group is abelian can be expressedas a sum of roots of unity with rational coecients. For example,

p5 = e2i/5 e4i/5 e6i/5 + e8i/5:

The theorem is named after Leopold Kronecker and Heinrich Martin Weber.

19.1 Field-theoretic formulation

The KroneckerWeber theorem can be stated in terms of elds and eld extensions. Precisely, the Kronecker-Webertheorem states: every nite abelian extension of the rational numbers Q is a subeld of a cyclotomic eld. That is,whenever an algebraic number eld has a Galois group over Q that is an abelian group, the eld is a subeld of a eldobtained by adjoining a root of unity to the rational numbers.For a given abelian extension K of Q there is a minimal cyclotomic eld that contains it. The theorem allows oneto dene the conductor of K as the smallest integer n such that K lies inside the eld generated by the n-th roots ofunity. For example the quadratic elds have as conductor the absolute value of their discriminant, a fact generalisedin class eld theory.

19.2 History

The theorem was rst stated by Kronecker (1853) though his argument was not complete for extensions of degree apower of 2. Weber (1886) published a proof, but this had some gaps and errors that were pointed out and correctedby Neumann (1981). The rst complete proof was given by Hilbert (1896).

19.3 Generalizations

Lubin and Tate (1965, 1966) proved the local KroneckerWeber theorem which states that any abelian extension ofa local eld can be constructed using cyclotomic extensions and LubinTate extensions. Hazewinkel (1975), Rosen(1981) and Lubin (1981) gave other proofs.Hilberts twelfth problem asks for generalizations of the KroneckerWeber theorem to base elds other than therational numbers, and asks for the analogues of the roots of unity for those elds.

34

19.4. REFERENCES 35

19.4 References Ghate, Eknath (2000), The Kronecker-Weber theorem, in Adhikari, S. D.; Katre, S. A.; Thakur, Dinesh,

Cyclotomic elds and related topics (Pune, 1999), Bhaskaracharya Pratishthana, Pune, pp. 135146, MR1802379

Greenberg, M. J. (1974). An Elementary Proof of the Kronecker-Weber Theorem. American MathematicalMonthly (The American Mathematical Monthly, Vol. 81, No. 6) 81 (6): 601607. doi:10.2307/2319208.JSTOR 2319208.

Hazewinkel, Michiel (1975), Local class eld theory is easy, Advances in Mathematics 18 (2): 148181,doi:10.1016/0001-8708(75)90156-5, ISSN 0001-8708, MR 0389858

Hilbert, David (1896), Ein neuer Beweis des Kroneckerschen Fundamentalsatzes ber Abelsche Zahlkr-per., Nachrichten der Gesellschaft der Wissenschaften zu Gttingen (in German): 2939

Kronecker, Leopold (1853), "ber die algebraisch ausbaren Gleichungen, Berlin K. Akad. Wiss. (in Ger-man): 365374, Collected works volume 4

Kronecker, Leopold (1877), "ber Abelsche Gleichungen, Berlin K. Akad. Wiss. (in German): 845851,Collected works volume 4

Lemmermeyer, Franz (2005), Kronecker-Weber via Stickelberger, Journal de thorie des nombres de Bor-deaux 17 (2): 555558, doi:10.5802/jtnb.507, ISSN 1246-7405, MR 2211307

Lubin, Jonathan (1981), The local Kronecker-Weber theorem, Transactions of the American MathematicalSociety 267 (1): 133138, doi:10.2307/1998574, ISSN 0002-9947, MR 621978

Lubin, Jonathan; Tate, John (1965), Formal complex multiplication in local elds, Annals of Mathematics.Second Series 81: 380387, ISSN 0003-486X, JSTOR 1970622, MR 0172878

Lubin, Jonathan; Tate, John (1966), Formal moduli for one-parameter formal Lie groups, Bulletin de laSocit Mathmatique de France 94: 4959, ISSN 0037-9484, MR 0238854

Neumann, Olaf (1981), Two proofs of the Kronecker-Weber theorem according to Kronecker, and Weber"",Journal fr die reine und angewandte Mathematik 323: 105126, doi:10.1515/crll.1981.323.105, ISSN 0075-4102, MR 611446

Rosen, Michael (1981), An elementary proof of the local Kronecker-Weber theorem, Transactions of theAmerican Mathematical Society 265 (2): 599605, doi:10.2307/1999753, ISSN 0002-9947, MR 610968

afarevi, I. R. (1951), A new proof of the Kronecker-Weber theorem, Trudy Mat. Inst. Steklov. (in Russian)38, Moscow: Izdat. Akad. Nauk SSSR, pp. 382387, MR 0049233 English translation in his CollectedMathematical Papers

Schappacher, Norbert (1998), On the history of Hilberts twelfth problem: a comedy of errors, Matriauxpour l'histoire des mathmatiques au XXe sicle (Nice, 1996), Smin. Congr. 3, Paris: Socit Mathmatiquede France, pp. 243273, ISBN 978-2-85629-065-1, MR 1640262

Weber, H. (1886), Theorie der Abelschen Zahlkrper,ActaMathematica (in German) 8: 193263, doi:10.1007/BF02417089,ISSN 0001-5962

Chapter 20

Laorgues theorem

In mathematics,Laorgues theorem, due to Laurent Laorgue, completes the Langlands program for general lineargroups over algebraic function elds, by giving a correspondence between automorphic forms on these groups andrepresentations of Galois groups.The Langlands conjectures were introduced by Langlands (1967, 1970) and describe a correspondence betweenrepresentations of the Weil group of an algebraic function eld and representations of algebraic groups over thefunction eld, generalizing class eld theory of function elds from abelian Galois groups to non-abelian Galoisgroups.

20.1 Langlands conjectures for GL1The Langlands conjectures for GL1(K) follow from (and are essentially equivalent to) class eld theory. More pre-cisely the Artin map gives a map from the idele class group to the abelianization of the Weil group.

20.2 Representations of the Weil group

20.3 Automorphic representations of GLn(F)The representations of GLn(F) appearing in the Langlands correspondence are automorphic representations.

20.4 Drinfelds theorem for GL2

20.5 Laorgues theorem for GLn(F)Here F is a global eld of some positive characteristic p, and is some prime not equal to p.Laorgues theorem states that there is a bijection between:

Equivalence classes of cuspidal representations of GLn(F), and Equivalence classes of irreducible -adic representations () of dimension n of the absolute Galois group of

F

that preserves the L-function at every place of F.The proof of Laorgues theorem involves constructing a representation () of the absolute Galois group for eachcuspidal representation . The idea of doing this is to look in the -adic cohomology of the moduli stack of shtukasof rank n that have compatible level N structures for all N. The cohomology contains subquotients of the form

36

20.6. APPLICATIONS 37

()()

which can be used to construct () from . A major problem is that the moduli stack is not of nite type, whichmeans that there are formidable technical diculties in studying its cohomology.

20.6 ApplicationsLaorgues theorem implies the RamanujanPetersson conjecture that if an automorphic form for GLn(F) has centralcharacter of nite order, then the corresponding Hecke eigenvalues at every unramied place have absolute value 1.Laorgues theorem implies the conjecture of Deligne (1980, 1.2.10) that an irreducible nite-dimensional l-adicrepresentation of the absolute Galois group with determinant character of nite order is pure of weight 0.

20.7 See also

Local Langlands conjectures

20.8 References

Borel, Armand (1979), Automorphic L-functions, in Borel, Armand; Casselman, W., Automorphic forms,representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 2761, ISBN978-0-8218-1437-6, MR 546608

Deligne, Pierre (1980), La conjecture de Weil. II, Publications Mathmatiques de l'IHS (52): 137252,ISSN 1618-1913, MR 601520

Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) [1966], Representation theory and automorphicfunctions, Generalized functions 6, Philadelphia, Pa.: W. B. Saunders Co., ISBN 978-0-12-279506-0, MR0220673

Laorgue, Laurent (1998), Proceedings of the International Congress of Mathematicians (Berlin, 1998)",Documenta Mathematica II: 563570, ISSN 1431-0635, MR 1648105 |chapter= ignored (help)

Laorgue, Laurent Chtoucas de Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands.(Drinfeld shtukas, Arthur-Selberg trace formula and Langlands correspondence) Proceedings of the Interna-tional Congress of Mathematicians, Vol. I (Beijing, 2002), 383400, Higher Ed. Press, Beijing, 2002.

Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics 114,Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, MR 0401654

Langlands, Robert (1967), Letter to Prof. Weil

Langlands, R. P. (1970), Problems in the theory of automorphic forms, Lectures in modern analysis and appli-cations, III, Lecture Notes in Math 170, Berlin, New York: Springer-Verlag, pp. 1861, doi:10.1007/BFb0079065,ISBN 978-3-540-05284-5, MR 0302614

Grard Laumon, The work of Laurent Laorgue Proceedings of the ICM, Beijing 2002, vol. 1, 9197

G. Laumon La correspondance de Langlands sur les corps de fonctions (d'apres Laurent Laorgue) (The Lang-lands correspondence over function elds (according to Laurent Laorgue)) Seminaire Bourbaki, 52eme an-nee, 19992000, no. 873

38 CHAPTER 20. LAFFORGUES THEOREM

20.9 External links Laorgues publications The work of Robert Langlands Rapoport, The work of Laurent Laorgue

Chapter 21

Landau prime ideal theorem

In algebraic number theory, the prime ideal theorem is the number eld generalization of the prime number theorem.It provides an asymptotic formula for counting the number of prime ideals of a number eld K, with norm at most X.

21.1 ExampleWhat to expect can be seen already for the Gaussian integers. There for any prime number p of the form 4n + 1, pfactors as a product of two Gaussian primes of norm p. Primes of the form 4n + 3 remain prime, giving a Gaussianprime of norm p2. Therefore we should estimate

2r(X) + r0(pX)

where r counts primes in the arithmetic progression 4n + 1, and r in the arithmetic progression 4n + 3. By thequantitative form of Dirichlets theorem on primes, each of r(Y) and r(Y) is asymptotically

Y

2 logY :

Therefore the 2r(X) term predominates, and is asymptotically

X

logX :

21.2 General number eldsThis general pattern holds for number elds in general, so that the prime ideal theorem is dominated by the ideals ofnorm a prime number. As Edmund Landau proved in Landau 1903, for norm at most X the same asymptotic formula

X

logXalways holds. Heuristically this is because the logarithmic derivative of the Dedekind zeta-function of K always hasa simple pole with residue 1 at s = 1.As with the Prime Number Theorem, a more precise estimate may be given in terms of the logarithmic integralfunction. The number of prime ideals of norm X is

Li(X) +OK(X exp(cKp

log(X));where cK is a constant depending on K.

39

40 CHAPTER 21. LANDAU PRIME IDEAL THEOREM

21.3 See also Abstract analytic number theory

21.4 References Alina Carmen Cojocaru; M. Ram Murty. An introduction to sieve methods and their applications. London

Mathematical Society Student Texts 66. Cambridge University Press. pp. 3538. ISBN 0-521-61275-6. Landau, Edmund (1903). Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes. Mathematische

Annalen 56 (4): 645670. doi:10.1007/BF01444310. Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge

tracts in advanced mathematics 97. pp. 266268. ISBN 0-521-84903-9.

Chapter 22

Local Tate duality

In Galois cohomology, local Tate duality (or simply local duality) is a duality for Galois modules for the absoluteGalois group of a non-archimedean local eld. It is named after John Tate who rst proved it. It shows that the dualof such a Galois module is the Tate twist of usual linear dual. This new dual is called the (local) Tate dual.Local duality combined with Tates local Euler characteristic formula provide a versatile set of tools for computingthe Galois cohomology of local elds.

22.1 StatementLet K be a non-archimedean local eld, let Ks denote a separable closure of K, and let GK = Gal(Ks/K) be the absoluteGalois group of K.

22.1.1 Case of nite modules