HOMEWORK 7

1. Let E be a subfield in F (X) properly containing F. Show that F (X) is algebraic over E.

2. Prove that any automorphism of ℝ is the identity. (Hint: Show that for any σ ∈ Gal(ℝ/ℚ), σ(a) ≥ a, a ∈ ℝ.)

3. Let K/F be a Galois extension, G = Gal(K/F). Prove that for every α, β ∈ K that have the same minimal polynomial over F there is σ ∈ G such that σ(α) = β.

4. Show that the fields Q(\sqrt{2}, \sqrt{3}) and Q(\sqrt{2}, \sqrt{5}) are not isomorphic.

5. Let K = ℚ(α) with α a root of X^{3} + X^{2} − 2X − 1 ∈ ℚ[X]. Show that K/ℚ is normal.

6. Let K = ℚ(\sqrt{2}, \sqrt{3}, α) where α^{2} = (9 − 5\sqrt{3})(2 − \sqrt{2}). Show that K/ℚ is normal and determine Gal(K/ℚ).

7. Suppose that K/F is Galois. Let F ⊂ E ⊂ K and L the smallest subfield of K containing E and such that L/F is normal. Prove that

Gal(K/L) = ⋂_{σ∈Gal(K/F)}σGal(K/E)σ^{−1}.

8. Let K/F be an extension of finite fields.

a) Prove that the norm map N : K → F is surjective.

b) Let F be a finite field. Prove that any element in F is a sum of two squares.

9. Let X^{3} + pX + q be an irreducible polynomial over a finite field F of characteristic 3. Show that −p is a square in F.

10. Let K be a subfield of ℝ, f an irreducible polynomial over K of degree 4. Suppose that f has exactly two real roots. Show that the Galois group of f is either S_{4} or of order 8.