f (x) ∈ F [x] is an irreducible polynomial. Show that f (x) has multiple ze- ros if and only if there exists g(x) ∈ F [x] such that f (x) = g(x p ).

1

PROBLEM SET 14 DUE: June 9 Problem 1

Let F be a finite field with q = p ⁿ elements, where p is a prime.

f (x) ∈ F [x] is an irreducible polynomial. Show that f (x) has multiple ze- ros if and only if there exists g(x) ∈ F [x] such that f (x) = g(x ^p ).

Problem 2

Let F be a finite field with q = p ⁿ elements, where p is a prime. Let H be a subgroup of Aut(F ) of order m. K = {a ∈ F |∀σ ∈ H, σ(a) = a}.

Prove that:

(1). m|n.

(2). K is the unique subfield of F of order p

.

Problem 3

Let Ω p be the algebraic closure of the finite field F p where p is a prime, and the unique subfield of Ω p of order p ⁿ is denoted by F p

. Show that:

(1). F p

⊂ F p

if and only if m|n.

(2). Suppose that F _p

⊂ F p

. Let G = {σ ∈ Aut(F _p

)|∀a ∈ F _p

, σ(a) = a}. Then G is a cyclic group of order _m ⁿ .

Problem 4

Let K be a field, σ ∈ Aut(K). Show that K/K ^σ is a separable exten- sion.

Problem 5

Show that for n ≤ m, there exist n matrices A 1 , A 2 , ...., A n ∈ M m×m (F q ) such that ∀(x 1 , x 2 , ...., x n ) 6= 0,we have

x 1 A 1 + x 2 A 2 + .... + x n A n 6= 0.

Problem 6

2

(1). Show that any functions f : F _q ⁿ → F q can be represented by an el- ement in F q [x 1 , ...., x n ]. That is, there exists a polynomial g ∈ F q [x 1 , ...., x n ] such that ∀x ∈ F _q ⁿ , g(x) = f (x).

*(2). For the given f as above, g ∈ F q [x 1 , ...., x n ] is uniquely deter- mined after modulo the ideal (x ^q ₁ − x 1 , ...., x ^q _n − x n ).

Problem 7

Let E = Q( √ 2, √

3, u), u ² = (9 − 5 √

3)(2 − √

2). Show that E/Q is a

Galois extension and determine its Galois group.