HOMEWORK 5

1. Determine the Galois group of the extension K = ℚ(\sqrt{2}, \sqrt{3}, \sqrt{5})/ℚ and all subfields of K.

2. Determine the Galois group of the extension ℚ(^{8}\sqrt{2}, i)/ℚ(\sqrt{−2}).

3. Determine the splitting field of the polynomial X^{4} − 2X^{2} − 2 over ℚ and its Galois group.

4. Determine the splitting field of the polynomial X^{6} +3 over ℚ and its Galois group.

5. Let F be a field. Prove that F (X, Y)/F (XY, X + Y) is a Galois extension and determine its Galois group.

6. Let F be a field of characteristic different from 2. Prove that F (X, Y)/F (X^{2}, XY, Y^{2}) is a Galois extension and determine its Galois group.

7. Let K/F be a Galois extension of degree p^{n} (p is prime, n > 0). Show that there is a subfield L ⊂ K such that L/F is a cyclic extension of degree p.

8. Let K/F be a Galois extension with Gal(K/F) = G. For any α ∈ K define the norm of α to be

N(α) = ∏_{σ∈G} σ(α).

a) Prove that N(α) ∈ F.

b) Prove that N(αβ) = N(α)N(β) for all α, β ∈ K.

c) Find the norm of α = a + b\sqrt{d} for the quadratic extension ℚ(\sqrt{d})/ℚ.

9. Let K/F be a Galois quadratic extension, α ∈ K. Prove that the following two conditions are equivalent:

(1) N(α) = 1.

(2) There exists β ∈ K such that

α = \frac{β}{σ(β)}

where σ is the nontrivial element in Gal(K/F). (Hint: Consider β = 1 + α.)

10. Let p be a prime integer. Let F be a field having no nontrivial extensions of degree prime to p. Prove that every separable extension K/F of degree p is Galois.