Math 567 Homework 8
Fall 2012
Due: Friday, October 26
1. Marcus, Exercise 5, (a)(c), page 115: Let L/K be an extension of number elds such
that L is Galois over K with Galois group G. Let p be a prime of K . By intermediate
eld we will mean intermediate e
Math 567 Homework 9
Fall 2012
Due: Friday, November 2
1. Marcus, Exercise 3b, page 114: Complete the proof of the quadratic reciprocity law.
(You may assume part (a).)
In particular, prove that the quadratic subeld of Q(e2i/p ) is Q( p) if p is prime and
Math 567 Homework 2
Fall 2012
Due: Friday, September 14
1. Let A be an integrally closed domain, and let K be its eld of fractions. Let f (X )
A[X ] be a monic polynomial. If f (X ) is reducible in K [X ], show that it is reducible
in A[X ].
2. Let K = Q
Math 567 Homework 1
Fall 2012
Due: Friday, September 7
1. Show that Z[ 5] is not integrally closed, and deduce that cannot be a unique facit
torization domain. Give an example of an element of Z[ 5] that has two distinct
factorizations into irreducible el
Math 567 Homework 10
Fall 2012
Due: Friday, November 16
1. (5 points) Marcus, Exercise 23, page 123: Let V1 be the ramication group dened in
Problem 4 on Homework 9, and use the same notation as in this problem. Show that
V1 is the Sylow p-subgroup of E ,
Math 567 Homework 11
Fall 2012
Due: Monday, December 3
1. (5 points) Prove that the ideal class group of Q[ 21] is the Klein four group.
2. (5 points) Let K := Q[ 1, 5]. Show that OK = Z[ 1, 1+2 5 ]. Show that the only
primes in Z that ramify in K are 2 a
Math 567 Homework 7
Fall 2012
Due: Friday, October 19
1. (5 points) Marcus, Exercise 1, page 114: Show that E (Q|p) is a normal subgroup of
D(Q|p) directly from the denition of these groups.
2. (10 points) Let m, n be distinct square-free integers = 1. Th
Math 567 Homework 12
Fall 2012
Due: Wednesday, December 12
1. (10 points) Marcus, Exercise 33, page 152.
2. (5 points) Let | | be a nonarchimedean absolute value on a eld K .
(a) Dene an open disk with radius r and center a to be
D(a, r) = cfw_x K : |x a|
Math 567 Homework 3
Fall 2012
Due: Friday, September 21
1. Show that the ring of integers OL of a number eld L is the largest subring of L that
is nitely generated as a Z-module.
2. For n > 1 an integer, prove that 2 cos(2/n) and 2 sin(2/n) are algebraic
Math 567 Homework 4
Fall 2012
Due: Friday, September 28
1. Let f (x) = x3 + ax + b with a, b Z, and assume f is irreducible over Q. Let be a
root of f .
(a) Show that f () = (2a + 3b)/.
(b) Show that 2a + 3b is a root of
Use this to nd
Q[]
NQ (2a +
3
x 3
Math 567 Homework 5
Fall 2012
Due: Friday, October 5
For this homework you may use the results in Marcus up to (and including) page 71.
1. Let K be any eld and let L be a nite separable extension of the eld K (x) of rational
functions over K . Prove that
Math 567 Homework 6
Fall 2012
Due: Friday, October 12
1. (5 points) Let m = 1 be a square-free integer. Let K := Q( m).
(a) List the possible factorizations of (p) in OK . Justify your answer.
(b) Show that if p | disc(K ), then (p) ramies in OK .
(c) Sho