Pries: M467 - Abstract Algebra I, Spring 2011
Homework 2: Field extensions and automorphisms
Be ready to present two problems by 2/25 and turn in six problems by 3/4.
Read: Judson 21.1, Ash 3.2, 6.1, Reid 3.3, Wilkons 4.4, 4.5, 4.8
1. Recap of HW 1 problems 2, 4, 6, 8, 9, 11
2. Find a field extension L of Q such that Aut(L : Q) ' Z/2 × Z/2 × Z/2. How many
quadratic fields K are contained in L?
3. Let ζ = ζ7 and let K = Q(ζ).
(a) Find an automorphism τ of K with order 6
(in other words, τ 6 = id but τ i 6= id if 1 ≤ i ≤ 5).
(b) What is the difference between τ 2 (ζ) and τ (ζ 2 )?
(c) For which i between 1 and 5, is τ i (ζ + ζ 6 ) = ζ + ζ 6 ?
What is Aut(K : Q(ζ + ζ 6 ))?
(d) For which i between 1 and 5, is τ i (ζ + ζ 2 + ζ 4 ) = ζ + ζ 2 + ζ 4 ?
What is Aut(K : Q(ζ + ζ 2 + ζ 4 ))?
4. Let α be a root of f (x) = x4 − 5. Let K = Q(α).
(a) Find Aut(K : Q).
(b) What is the splitting field L of f (x)? Find a basis for L over Q.
(c) Show that Aut(L : Q) ' D4 .
(d) What is the fixed fields of (r) and (r2 )?
(e) What are the fixed fields of (f ), r2 f , rf , and r3 f ?