Math 901
Homework # 3
Due: Friday, October 2nd
1. Let E be the splitting field of f (x) = x6 + 3 ∈ Q[x] and let α ∈ E be a root of f (x).
(a) [10] Prove that E = Q(α). (Hint: Note that
unity.)
√
1+ 3i
2
is a primitive 6th root of
(b) [10] Let G be the Galois group of E/Q. Determine, with justification, whether
G is abelian.
2. Let E/F be a finite Galois field extension with Galois group G. Let α ∈ E and H the
Galois group of E/F (α). Let σ1 , . . . , σn be a complete set of coset representatives for
H in G. (I.e., n = [G : H] and σi H 6= σj H for all i 6= j.) Prove that the minimal
polynomial of α over F is
n
Y
(x − σi (α)).
i=1
3. Let E be the splitting field of x4 − 2 over Q. Find a presentation (with justification)
for Gal(E/Q).
4. Let E/F be a finite Galois extension and K an intermediate field. Let G = Gal(E/F )
and H = Gal(E/K). Prove that NG (H) = {g ∈ G | g(K) = K} and NG (H)/H ∼
=
Aut(K/F ).
5. Let K ⊆ E, F ⊆ L are fields and suppose E/K is finite and Galois. Prove that EF/F
is Galois and Gal(EF/F ) is isomorphic to a subgroup of Gal(E/K).
6. Let F be a field and f (x) ∈ F [x] a separable irreducible polynomial of prime degree.
Let α be a root of f (x) and suppose f (x) has at least two roots in E = F (α). Prove
that E is the splitting field for f (x) and that E/F is cyclic. (Hint: Let L be the normal
closure of E/F , G = Gal(L/F ), and H = Gal(L/E). Using problem # 4, prove that
H 6= NG (H).)
E
7. Let E be a finite extension of a finite field F . Prove that TrE
F and NF are surjective
(as maps from E to F ). (Recall from Math 818 that E/F is a cylic extension.)