mlbaker.org presents
PMATH 442/642
Fields and Galois Theory
Dr. John Lawrence Fall 2012 (1129) University of Waterloo
Disclaimer: These notes are provided as-is, and may be incomplete or contain errors.
Contents
1 Review
2
2 Groups
3
3 Rings
6
4 Fields
8

PMATH 442, Galois Theory
Assignment 5, due November 23
1. Prove that if K/F is a nite extension and S is the set of intermediate elds, then |S| |F |
or |S| < (so if F is uncountable either S is uncountable or nite).
2. Let F be a nite eld. Prove that if x

PMATH 442, Galois Theory
Assignment 4, due November 9
1. Prove that R(t)/R is a Galois extension (R(t) is the rational function eld).
2. Let 24 be a 24th root of unity over Q.
(a) Describe Gal(Q(24 )/Q) up to isomorphism.
(b) Describe Gal(Q(24 )/Q(i ) up

PMATH 442, Galois Theory
Assignment 3, due October 24
1. (a) Prove that Z7 (t 9 ) Z7 (t) is not a normal extension (Z7 (t) is the rational function
eld).
(b) Conclude that L/F and K/L normal does not imply (in general) that K/F is normal.
2. Let F be a el

PMATH 442, Galois Theory
Assignment 1, due September 28
1. (a) Prove that if A and B are solvable normal subgroups of G , then so is AB = cfw_ab : a
A, b B .
(b) Conclude that a nite group has a unique maximal solvable normal subgroup.
2. A subgroup G of

PMATH 442, Galois Theory
Assignment 2, due October 12
1. (a) Let p and q be distinct (positive) primes in Z. Prove that [Q( p, q) : Q] = 4.
5
2+3
:Q .
(b) Calculate Q
1 52
2. Let F (x, y ) be the rational function eld in two indeterminates. Prove that