MATH 361: NUMBER THEORY TWELFTH LECTURE
Let
1 + 3
.
2
The subjects of this lecture are the arithmetic of the ring
= 3 = e2i/3 =
D = Z[ ],
and the cubic reciprocity law.
1. Unique Factorization
The ring D is Euclidean with norm function
N : D Z0 ,
N (a +
MATH 361: NUMBER THEORY ELEVENTH LECTURE
The subjects of this lecture are characters, Gauss sums, Jacobi sums, and counting formulas for polynomial equations over nite elds.
1. Definitions, Basic Properties
Let p be an odd prime. (However, essentially eve
MATH 361: NUMBER THEORY NINTH LECTURE
1. Algebraic numbers and algebraic integers
We like numbers such as i and = 3 = e2i/3 and (1 + 5)/2 and so on. To
think about such numbers in a structured way is to think of them not as radicals,
but as roots.
Denitio
MATH 361: NUMBER THEORY EIGHTH LECTURE
1. Quadratic Reciprocity: Introduction
Quadratic reciprocity is the rst result of modern number theory. Lagrange
conjectured it in the late 1700s, but it was rst proven by Gauss in 1796. From a
naive viewpoint, there
MATH 361: NUMBER THEORY TENTH LECTURE
The subject of this lecture is nite elds.
1. Root Fields
Let k be any eld, and let f (X ) k[X ] be irreducible and have positive degree.
We want to construct a supereld K of k in which f has a root. To do so, consider
MATH 361: NUMBER THEORY SEVENTH LECTURE
1. The Unit Group of Z/nZ
Consider a nonunit positive integer,
n=
pep > 1.
The Sun Ze Theorem gives a ring isomorphism,
Z/nZ
=
Z/pep Z.
The right side is the cartesian product of the rings Z/pep Z, meaning that add
MATH 361: NUMBER THEORY FOURTH LECTURE
1. Introduction
Everybody is familiar with modular arithmetic, meaning the usual arithmetic of
the integers subject to the additional condition that some xed integer (such as 12)
is to be viewed as 0. Three hours aft
MATH 361: NUMBER THEORY SIXTH LECTURE
Let d be a positive integer. Consider a polynomial in d variables with integer
coecients,
call
f Z[X1 , , Xd ] = Z[X ].
Consider also a succession of conditions, each stronger than the next:
(A) The equation f (X ) =
MATH 361: NUMBER THEORY THIRD LECTURE
1. Introduction
The topic of this lecture is arithmetic functions and Dirichlet series.
By way of introduction, consider Euclids proof that there exist innitely many
primes: If p1 through pn are prime then the number
MATH 361: NUMBER THEORY SECOND LECTURE
1. Introduction
The topic of this lecture is eventually the unique factorization theorem for the
integers:
Theorem 1.1. Let n be a nonzero integer. Then n factors as
n = pe1 per ,
r
1
r 0, p1 , , pr P , e1 , , er Z+
MATH 361: NUMBER THEORY FIRST LECTURE
1. Introduction
As a provisional denition, view number theory as the study of the properties of
the positive integers,
Z+ = cfw_1, 2, 3, .
Of particular interest, consider the prime numbers, the noninvertible positive
MATH 361: NUMBER THEORY FIFTH LECTURE
1. The Sun Ze Theorem
The Sun Ze Theorem is often called the Chinese Remainder Theorem. Here is
an example to motivate it. Suppose that we want to solve the equation
13x = 23 mod 2310.
(Note that 2310 = 2 3 5 7 11.) S