ALGEBRA HANDOUT 1: RINGS, FIELDS AND GROUPS
PETE L. CLARK
1. Rings
Recall that a binary operation on a set S is just a function : S S S : in other
words, given any two elements s1 , s2 of S , there is a well-dened element s1 s2 of S .
A ring is a set R en
ALGEBRA HANDOUT 2.5: MORE ON COMMUTATIVE
GROUPS
PETE L. CLARK
1. Reminder on quotient groups
Let G be a group and H a subgroup of G. We have seen that the left cosets xH
of H in G give a partition of G. Motivated by the case of quotients of rings by
ideal
PYTHAGOREAN TRIPLES
PETE L. CLARK
1. Parameterization of Pythagorean Triples
1.1. Introduction to Pythagorean triples.
By a Pythagorean triple we mean an ordered triple (x, y, z ) Z3 such that
x2 + y 2 = z 2 .
The name comes from elementary geometry: if a
GAUSSS CIRCLE PROBLEM
1. Introduction
We wish to study a very classical problem: how many lattice points lie on or inside
the circle x2 + y 2 = r2 ? Equivalently, for how many pairs (x, y ) Z2 do we have
x2 + y 2 r2 ? Let L(r) denote the number of such pa
4400/6400 PROBLEM SET 0
0) Proof of Fermats Last Theorem, Step 0: For a positive integer n, let
F LT (n) denote the following statement: for x, y, z Z such that xn + y n = z n ,
xyz = 0. Show that if FLT(4) holds and FLT(p) holds for each odd prime p, the
4400/6400 PROBLEM SET 1
Key: (E) denotes easy. If you honestly feel the problem is too easy, just write
okay, but try to solve some harder problems as well.
1.1)(E) Prove the Division Theorem: If a b > 0 are integers, then there exist unique non-negative
4400/6400 PROBLEM SET 2
A sucient number of problems: 6 for 4400 students, 8 for 6400 students.
The rst four problems pertain to the Euclidean Algorithm, which will be applied
to positive integers a b 1.
2.1) Explain how to use a nonprogrammable handheld
4400/6400 PROBLEM SET 3
A sucient number of problems: 4 for 4400 students, 6 for 6400 students.
3.1) Let c and N > 1 be integers, and let c be the class of c modulo N .
a) Show that c is a unit in Z/N Z if and only if gcd(c, N ) = 1.
b) Show that #(Z/N Z)
4400/6400 PROBLEM SET 4
A sucient number of problems: 5 for 4400 students, 8 for 6400 students.
4.1) Determine exactly which integers n, 1 n 100, are sums of two squares.
4.2) a) The rst run of two consecutive non-negative integers which are not
sums of t
4400/6400 PROBLEM SET 5
PETE L. CLARK
A sucient number of problems: 4 for 4400 students, 5 for 6400 students.
5.1) Evaluate these Legendre symbols (the denominators are all prime numbers):
85
101
,
29
241
,
101
1987
,
31706
43789
.
5.2) Make up another si
4400/6400 PROBLEM SET 6
Recommendation: 4400 students should do at least three problems; 6400 students
should do at least four.
Heads up: There is an online applet for nding the fundamental solution to a
Pell equation, available at
http:/www.numbertheory.
4400/6400 PROBLEM SET 7
Recommendation: 4400 students should do at least 4 problems; 6400 students
should do at least 6.
7.1) Write out a careful proof of Lemma 6 in the Minkowskis Theorem handout.
Especially, say a bit about the change-of-volume properti
QUADRATIC RECIPROCITY II: THE PROOFS
PETE L. CLARK
We shall prove the Quadratic Reciprocity Law and its second supplement.
1. Preliminaries on congruences in cyclotomic rings
For a positive integer n, let n = e
2i
n
be a primitive nth root of unity, and l
Quaternion Algebras and Quadratic Forms
by
Zi Yang Sham
A thesis
presented to the University of Waterloo
in fulllment of the
thesis requirement for the degree of
Master of Mathematics
in
Pure Mathematics
Waterloo, Ontario, Canada, 2008
c Zi Yang Sham 2008
QUADRATIC RINGS
PETE L. CLARK
1. Quadratic fields and quadratic rings
Let D be a squarefree integer not equal to 0 or 1. Then D is irrational, and
Q[ D], the subring of C obtained by adjoining D to Q, is a eld.
From an abstract algebraic perspective, an e
QUADRATIC RECIPROCITY I
PETE L. CLARK
We now come to the most important result in our course: the law of quadratic
reciprocity, or, as Gauss called it, the aureum theorema (golden theorem).
Many beginning students of number theory have a hard time appreci
SOME IDEAS FOR FINAL PROJECTS
PETE L. CLARK
1. Project ideas
1.1. Write a computer program that plays Schuhs divisor game better
than you (or I) do.
1.2. Nonunique factorization in the ring R[cos , sin ] of real trigonometric
polynomials.
Reference: H.F.
INTEGRAL ELEMENTS AND EXTENSIONS
PETE L. CLARK
Recall that a complex number is said to be an algebraic integer if is the root
of a nonconstant monic polynomial with Z coecients: i.e., if there exists an n and
integers a0 , . . . , an1 such that
n + an1 n1
ARITHMETICAL FUNCTIONS I: MULTIPLICATIVE
FUNCTIONS
PETE L. CLARK
1. Arithmetical Functions
Denition: An arithmetical function is a function f : Z+ C.
Truth be told, this denition is a bit embarrassing. It would mean that taking any
function from calculus
ARITHMETICAL FUNCTIONS II: CONVOLUTION AND
INVERSION
PETE L. CLARK
1. Sums over divisors, convolution and Mobius Inversion
The proof of the multiplicativity of the functions k , easy though it was, actually establishes a more general result. Namely, suppo
ARITHMETICAL FUNCTIONS III: ORDERS OF MAGNITUDE
1. Introduction
Having entertained ourselves with some of the more elementary and then the more
combinatorial/algebraic aspects of arithmetical functions, we now grapple with
what is fundamentally an analyti
THE CHEVALLEY-WARNING THEOREM (FEATURING. . . THE
ERDOS-GINZBURG-ZIV THEOREM)
PETE L. CLARK
1. The Chevalley-Warning Theorem
In this handout we shall discuss a result that was conjectured by Emil Artin in
1935 and proved shortly thereafter by Claude Cheva
DIRICHLET SERIES
PETE L. CLARK
1. Introduction
In considering the arithmetical functions f : N C as a ring under pointwise
addition and convolution:
f g (n) =
f (d1 )g (d2 ),
d1 d2 =n
we employed that old dirty trick of abstract algebra. Namely, we introd
DIRICHLETS THEOREM ON PRIMES IN ARITHMETIC
PROGRESSIONS
PETE L. CLARK
1. Statement of Dirichlets theorem
The aim of this section is to give a complete proof of the following result:
Theorem 1. (Dirichlet, 1837) Let a, N Z+ be such that gcd(a, N ) = 1. The
FOUNDATIONS AND THE FUNDAMENTAL THEOREM
PETE L. CLARK
1. Foundations
What is number theory?
This is a dicult question to answer: number theory is an area, or collection
of areas, of pure mathematics that have been studied for well over two thousand
years.
SOME IRRATIONAL NUMBERS
PETE L. CLARK
Proposition 1. The square root of 2 is irrational.
Proof. Suppose not: then there exist integers a and b = 0 such that 2 = a ,
b
2
meaning that 2 = a2 . We may assume that a and b have no common divisor if
b
they do,
A THEOREM OF MINKOWSKI; THE FOUR SQUARES
THEOREM
PETE L. CLARK
1. Minkowkskis Convex Body Theorem
1.1. Introduction.
We have already considered instances of the following type of problem: given a
bounded subset of Euclidean space RN , to determine #( ZN )
THE PELL EQUATION
1. Introduction
Let d be a nonzero integer. We wish to nd all integer solutions (x, y ) to
x2 dy 2 = 1.
(1)
1.1. History.
Leonhard Euler called (1) Pells Equation after the English mathematician John
Pell (1611-1685). This terminology ha
THE PRIME NUMBER THEOREM AND THE RIEMANN
HYPOTHESIS
PETE L. CLARK
1. Some history of the prime number theorem
Recall we have dened, for positive real x,
(x) = # cfw_primes p x.
The following is probably the single most important result in number theory.
THE PRIMES: INFINITUDE, DENSITY AND SUBSTANCE
PETE L. CLARK
1. There are infinitely many primes
The title of this section is surely, along with the uniqueness of factorization, the
most basic and important fact in number theory. The rst recorded proof was