Roots of Unity

III : nth root extensions

( and ther Galois groups )

buttingused roots of unity to study Fpalzp( since Epa is splitting field for xpn

- l- y )

Newsted ① what is the Galois groupof

an extension at F that

arises by adjoining aroot

of unity to F ?② Understand extensions of F

that arise by adjoining some

" nth root"

lie,

aroot of

×"- c for some CE F)

Dein A root of unity w is called a primitiventh root of unity it Ld

-- n

.

Mutt .order

Okami w is a primitive nth root of unity iff

{w, uh, . . ., w""

,t) are all district iff all

solution,

to x"- I are precisely {w, w} . . ,

w""

,1) (w/o repeats)

No - I is not a primitive 4th root of

unity since f- 1)2=1 (t is a prim 2nd

root of unity iff chartisn't 2)

When do primitive nth mob of unity exist ?

a gaunt F will be "missing" lots of

primitive nth mob at unity

therm ( when primitive nth mob of unity exist)

Thee exists some primitive nth not of unity

in an some extension Elfiff chart F) X n .

Pt (sketch) Boils down to whether x"- l is

separable , which we measure with god (x"- I , nxn" )pg

theorem ( Galois group ofneat - of - unity extension)

let E -- F ( un ) where Wn is a primitive nth root

of unity . Then Gall EIF ) U ( kn) ." injection

" Tiggsof kn

(Recall from Math 305 : Utkn) -- { Os Ken : gcdlk,n)

PI let reGall EIF) .

Recall from video 1 :

r is determined by the value of r( w.) ." Bob the Builder

"

says .

. since Wan - I -- O,we get

wn is a not at x" - I,Then we get

WoHwa) is also a root of x

"- I

. z

But roots at X"- I are precisely {wn.wii.in", 11

,

so we get r ( Wn) = wink fer some OE Kan .

Since r is an isomorphism , it preservesorder

,so

n-

- ( Wnt -

-lolwntl-lw.nl/--gcdTn.k7HeucegcdCn,k)=I,ie keµq¥Muth

Define 4 : Gullett ) -sulk. ) by

4G) ="

log w. ( rlwn ))"

( ie , k from lastslide)

OpPraruiy? bet r, I c-Gal (EIF) be givin .

let 4G) : i ( so rlwn): wni )

4(F) = j ( so Flwn )-

- wit).

Want Hrt ) -

- 4G) yet) = ij lmodn) .

Check : 4kt) determined by ritual) -- Hwa's )=o(wn)'

= wnij ①

Injectionsuppose r,F

C- Gal LEIF) and 463=48) -- i

.

Want to show: r -

- F.

observe That 4k) -- i means rlwn) -- wi'

and likewise YLF) : imeans F ( wa) = wni .

Since an element in Gal LEIF)is determined

by its action on wn,

and since rlwntolwn?

we get E- F . IDK,

Ey ht p>2 be prime . Ask : what is Gul (QlwpYa )

where wpis a primitive ph root of unity

in some extension of . ( e.g. , wp could

be cast It i sent ) EE )

From the last result : Gnlialwpllo )↳ Utkp)

On the other hand, Qlwp)/Q is the Kp#

splitting field for Epix) t Ix)un

Galo#mic pdga.m.in ,why ?

Wpis a primitive pin root of unity means

roots at XP - I are { up ,wp'

, up",1)

"

(x- 1) ( xp-' txt'-'t--txt l )

t¥Hence ( Gull (up)/Q ) Is [ Qlwp) : Q )

= 2(irralwp ))-

- 2 ( Ep (x) ) =p- l .

Since ( Gallup I - p - I = Nkp)

and Suica foul Cwp)/Q ) -7 Nkp),we get Gull Cwp)/a) = Nkp ) .

what happens if we adjoin roots at some

x"- c c- Fled to F ?

theorem ( Kummer Theory ,Part I) |

Suppose F carting a primitive ath pot of unity wn .

If the splitting field for flx) : xn -④c- Flex)

is E IF,Then Gal LEIF )→ In .

Furthermore, This is a surjection iff Hx) is ironed .

PI let * e E be some root of x" - c .Then all

roots of x"- c are precisely {a , wax , win, -- ion'd )

Bob the builder says that rt) is a root

of x"- c,

and hence Ma) = conks for som

OE Ken .

Define y '. Gul ( EIF ) → In by

4G) -"

logon ( old) - a" ) " ( ie

,The K above )

In Since E -- Fla )

, ayelement in Gal ( EIF) is

determined by it actin on a .

If 461=48) , then

T and I have the same action an x :

if ylo) - i. Then Ha)-- wind

if YLF) -- i , Then Fla)= wnia .

Optus let r,EEGAICEIF) . let ycr) - i. YCE) .

Heute da) - Wnit and FG) - Wix . Hence

←F)Ca) : r ( Fla)) - r( win ) = own)'r La)

-- up

'

. wni a= Wnit's 2 .

So Y( of) -

- it;

So we know : Gal ( EIF ) ↳ En.

We have left to show 4 is svrj iff th) is

irreducible .

Suppose fist that 4 is surjective . Hence for all

Oekcn,There exists some re Gal ( EIF )

with

Ha) : waka .

Now if irrpk) was a

proper factorat x

"-c = Hx)

,

then Bob the

Builder says : elements offed LEIF ) have

r la) goes toa root of irrp K ) .

we've already seen that ter ay Oskar,

there exists rt Gal LEIF) with da ) -- waka .

Hence irrp K) has n roots.But Then

Hire.sk )) en , and so in, la ) can't be

a proper divisorof x

"-c .

For other direction , uselemma 50 to pave : fer

ay OEKan,we can

find ofGal ( Elf ) so

old) -- waka .Hence Tlr) -- K . So Ysurjects. xx,

Cory ( Kummer They part I for primes )

let pbe prime , and assume Wp EF .

Then

for c E F#

and E The splitting held fer

XP - c,we

have

Gal (EIF ) = { Eide 3if cef

# P

Kp if c¢ IF# P

PI Ep has only side ) and Kp as subgroups .

Final step is to connect irreducibility of XP - c to

whether c is a pth power in F .

IDK,