HOMEWORK 6

1. Prove that a normal closure of a separable field extension is also separable (and hence Galois).

2. Let F be the finite field 𝔽_{q} and let n be prime to q. Show that the degree of the cyclotomic extension F (ξ_{n})/F is equal to the order of q + nℤ in (ℤ/nℤ)^{×}.

3. Find the cyclotomic polynomial Φ_{24}.

4. Determine all roots of unity in ℚ(\sqrt{−3}).

5. Find all n ≤ 10 such that ℚ(ξ_{n})/ℚ is a cyclic extension.

6. Find the Galois group of X^{4} − a over ℚ if

a) a = 1;

b) a = −1;

c) a = 2;

d) a = −2.

7. Find a splitting field L of X^{4} − 4X^{2} − 1 over ℚ and describe all subfields of L.

8. Let p be a prime integer and let F be a filed of characteristic different from p and containing a primitive p-th root of unity, a ∈ F^{×}. Prove that X^{p} − a is either irreducible or split over F.

9. Let p be a prime integer and let F be a field which has no nontrivial field extensions of degree not divisible by p. Show that for any finite separable field extension L/F there is a tower of fields F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} = L such that for any i = 0, 1, …, n − 1, F_{i+1}/F_{i} is a cyclic extension of degree p.

10. Let E/F be a Galois extension and let L and K be two intermediate fields such that L/F is normal. Assume that E = LK and L ∩ K = F. Show that Gal(E/F) is the semidirect product Gal(E/L) ⋊ Gal(E/K).