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GALOIS THEORY:

PROBLEM SHEET 4

(1) Let F be a finite field, show that the group F is cyclic. Deduce that F is a simpleextension ofFp.

(2) If is the Frobenius automorphism on Fpr , we may regard as an Fp-linear endomorphismof the Fp-vector space Fpr . So, forexample, we may think of as an rr matrix with entrieisin Fp (after picking a basis ofFpr). In HW 2, we have shown that the minimal p olynomialof is xr = 1.

(i) Determine the rational canonical form of .

(ii) Regarding as an r r-matrix over Fp, determine the Jordan canonical form of . Inparticular, give necessary and sufficient conditions for to be diagonalizable.

(3) This exercise shows that Aut(R/Q) is trivial!

(i) Show that any Aut(R/Q) takes positive reals to positive reals and conclude thata < b implies (a) < (b).

(ii) Deduce from (i) that is a continuous map from R to itself.

(iii) Deduce from (ii) that Aut(R/Q) is trivial.

(4i) Show that the automorphisms ofF(t) (the field of rational functions over a field F) whichare identity on F are of the form

f(t) f(at + bct + d

)

for a,b,c,d F and ad bc = 0. In other words, there is a surjective group homomorphismGL2(F) Aut(F(t)/F).

What is the kernel of this homomorphism?

(ii) What is the subfield of k(t) fixed by the automorphism which sends t to t + 1?

(5) Let K/F be a Galois extension with Galois group Gal(K/F). Let f(x) F[x] be anirreducible polynomial over F. We saw in class that each element of Gal(K/F) permutes theroots of f(x). Show that Gal(K/F) acts transitively on the set of roots of f(x).

(6i) Show that K =Q

(

2 +

2) is a Galois extension of

Qand determine Gal(K/

Q).

(ii) Determine the Galois group of the splitting field of x4 14x2 + 9 over Q.(7i) If f(x) F[x] is a separable polynomial with splitting field K and roots i K, showthat the element

disc(f) =

i

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2 GALOIS THEORY: PROBLEM SHEET 4

(ii) If f has degree 3, we showed in class that Gal(K/F) = A3 or S3. Show thatGal(K/F) = S3 if and only ifdisc(f) is not a square in F.

(8) The field R is obtained from the field Q by the process of completion with respect to themetric d(x, y) = |x y|. This procedure can be generalized as follows.(a) Let F be a field. A function | | : F R is a norm if

|x| 0 for all x F, with equality if and only if x = 0. |xy| = |x| |y|. |x + y| |x| + |y|.

Given a norm on F, define a metric d on F by setting

d(x, y) = |x y|.Show that the addition and multiplication maps on F are continuous functions.

(b) Let F denote the completion of F with respect to d. Show that the multiplication and

addition maps on F extend continuously to multiplication and addition maps on F. Withrespect to these operations, show that F is a field containing F as a dense subfield.

(c) As an example of this construction, we take F = Q, and for each prime p, we define thep-adic norm | |p on Q by:

|0|p = 0 and |x|p = |pa mn

|p = pa

for x = pam/n with m and n not divisible by p. Show that | |p is a norm. Indeed, showthat

|x + y|p max{|x|p, |y|p}.A norm satisfying this stronger condition is called a non-archimedean norm.

(d) Show that Q is not complete with respect to the metrix dp associated to | |p. Thecompletion of Q with respect to | |p is denoted by Qp and is called the field of p-adicnumbers.