Pries: M467 - Abstract Algebra I, Spring 2011
Homework 3: Galois groups
1. Let ζ = ζ11 and let L = Q(ζ). We know that G = Aut(L : Q) ' Z/10.
(a) Find an automorphism σ of order 2.
(b) Find an automorphism τ of order 5.
(c) Explain why στ has order 10.
2. Continuing problem 1, let H1 = hσi be the subgroup of size 2 in G.
(a) Find the fixed field E1 = LH1 .
(b) Find the minimal polynomial of ζ over LH1 .
(c) Find a basis of L over E1 and show that the degree [L : LH1 ] = 2.
(d) Match up the numbers 1, 2, 3, 4, 5 with A, B, C, D, E so that the following set S
is a basis for LH1 over Q:
S = {ζ A + ζ 6 , ζ B + ζ 7 , ζ C + ζ 8 , ζ D + ζ 9 , ζ E + ζ 10 }.
3. Continuing problem 2:
(a) Explain why there is a unique field E2 such that Q ⊂ E2 ⊂ L and such that
deg(E2 : Q) = 2.
(b) Find the squares mod 11. In other words, find the 5 numbers s1 , s2 , s3 , s4 , s5
between 1 and 10 so that si = x2 mod 11 for some 1 ≤ x ≤ 10.
(c) Let z = ζ s1 + ζ s2 + ζ s3 + ζ s4 + ζ s5 . Show that τ (z) = z.
(d) Show that Aut(Q(ζ) : Q(z)) = hτ i.
(e)* Find the minimal polynomial for z over Q. It has degree 2.