King Abdullah University of Science and Technology
CS 212

Spring 2014
% script for Problem 7.3
n = 2500; m = 250;
t = (1:n)'/8192; % 8192Hz: Matlab's default audio sampling rate
y = (sin(2*pi*521*t) + sin(2*pi*1233*t)/2; % synthesize a signal
D = dct(eye(n); % Transform matrix
k = randperm(n)'; k = sort(k(1:m); % Pick m
King Abdullah University of Science and Technology
CS 212

Spring 2014
ALGEBRAIC NUMBER THEORY
PMATH 441/641 ASSIGNMENT 2
Due in class on February 1st
Problem 1. Let A be an integral domain, and let a, b, c be ideals of A. Show that
a(b + c) = ab + ac.
Problem 2. Let Fp be the eld with p (prime) elements, and let Fp be its a
King Abdullah University of Science and Technology
CS 212

Spring 2014
ALGEBRAIC NUMBER THEORY
PMATH 441/641 ASSIGNMENT 3
Due in class on February 22nd
Problem 1. The point of this exercise is to show that there are no solutions to
the equation Y 2 = X 3 2 in Z, using an approach similar to that used in the notes
for solving
King Abdullah University of Science and Technology
CS 212

Spring 2014
ALGEBRAIC NUMBER THEORY
PMATH 441/641 ASSIGNMENT 1
Due in class on January 18th
Problem 1. Exercise 1.10 from the notes.
Problem 2. Exercise 1.15 from the notes.
Problem 3. Exercise 1.26 from the notes. Correction: the problem should say
Every PID which i
King Abdullah University of Science and Technology
CS 212

Spring 2014
ALGEBRAIC NUMBER THEORY
PMATH 441/641 ASSIGNMENT 5
Due in class on April 5th
Problem 1. Problem 8.9 in the notes.
Problem 2. Problem 8.10 in the notes.
Problem 3. Problems 8.12 and 8.13 in the notes.
Problem 4. Problem 8.14 in the notes.
Problem 5. Let K
King Abdullah University of Science and Technology
CS 212

Spring 2014
ALGEBRAIC NUMBER THEORY
PMATH 441/641 ASSIGNMENT 4
Due in class on March 15th
Problem 1. Suppose that f (X ) = X 3 + AX + B Z[X ] is irreducible, let f () = 0,
and let K = Q(). Compute DiscK/Q (1, , 2 ).
Problem 2. Let K be a eld, and let f (X ), g (X ) K
King Abdullah University of Science and Technology
CS 212

Spring 2014
Intro duction
0.1
Motivation
The development of algebraic number theory has been driven largely by a
single claim, the famous Last Theorem of Fermat.
Theorem 0.1 (Fermats Last Theorem). Let n 3 be an integer, and suppose that
xn + y n = z n ,
(1)
for some
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 8
The ideal class group
8.1
The ideal class group of a Dedekind domain
Weve seen that PIDs are much easier to work in than general rings.
Denition 8.1. Let A be a Dedekind domain. The ideal class group of
A, denoted C (A), is the quotient of the grou
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 10
Fermats Last Theorem
We are nally in a position to prove a special case of Fermats Last Theorem.
Theorem 10.1. Let q 11 be a prime, and let = e2i/q . There are no
solutions to the equation
xq + y q = z q
in the integers, unless q  xy z or q  hK
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 4
Linear algebra
Linear algebra is useful in considering nite extensions of elds, since the
larger eld is a nitedimensional vector space over the smaller. It turns out
that much of this analysis works for modules as well. The main goal of this
segme
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 5
Factoring primes in quadratic
extensions
5.1
Primes in quadratic extensions
Let A be a Dedekind domain with fraction eld K , let L be a nite extension
of K , and let B be the integral closure of A in L. We know that if p B is
a prime ideal, then q
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 2
Algebraic numb ers
2.1
The Chinese Remainder Theorem
Before proceeding with the material on algebraic numbers, well take a short
detour to present the Chinese Remainder Theorem.
Theorem 2.1. Let R be a ring, and let a and b be relatively prime idea
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 3
Dedekind domains
Before continuing with our exploration of number theory in rings, we need
some more background material from commutative algebra.
3.1
Mo dules
The study of extensions of elds is aided a great deal by considering a eld
as a vector s
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 6
Other elds
6.1
Factoring primes in general numb er elds
Before looking at the next special case, which is that of cyclotomic elds, we
will present the general version of Theorem 5.6. The following theorem says,
if OK = Z[], that the factorization o
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 7
Cyclotomic elds
7.1
Fermats Last Theorem
Before proceeding with an analysis of primes in cyclotomic elds, lets review
why this is interesting. Fermats Last Theorem is the claim that
xn + y n = z n
has no solutions in Z, for n 3, except the solution
King Abdullah University of Science and Technology
CS 212

Spring 2014
Week 9
The ideal class group
9.1
Lattices in Rn
Let 1 , ., n : K C be the embeddings of K into C, ordered so that
1 , ., r1 are the real embeddings, and r1 +1 , ., n are complex embeddings,
with r1 +r2 +j = r1 +j . We dene the canonical embedding of K int