We all know how indispensable it is to extend R
. Often it is better to start with Q
and make a finite extension, to get a number field
, but there are many more ways to do this-the Galois group of the algebraic closure of Q
is immensely complicated. The elements of a number field are algebraic numbers
, satisfying monic polynomial equations with coefficients in Q
. Among these are the algebraic integers, satisfying monic polynomial equations with coefficients in Z
, and they form a subring, the ring of integers
. When the number field is Q
, this subring is just Z
This course is about number fields and especially their rings of integers. In general these are not unique factorisation domains, but we shall see how unique factorisation can be restored by using ideals rather than elements. We are led naturally to consideration of the ideal class group and the unit group. This is not just a branch of algebra. We shall use also the geometry of numbers, and some
analytic functions, recognising the fact that an algebraic number can be thought of as an element of C
, and has a size.
Elementary number theory (primes, linear congruences).
Rings and groups, including irreducibles, units, Euclidean domains, quotient groups, First Isomorphism Theorem, and preferably the quotient of a ring by an ideal, and finitely generated abelian groups.
Field extensions, including the degree of an extension, irreducible polynomials, and preferably Eisenstein's criterion. Galois theory may be mentioned in passing, or in an exercise, but is not essential.
Finite extensions of Q
. Norms, traces and discriminants.
Proof that the subset of algebraic integers is a subring.
Existence of an integral basis for the ring of integers.
Ideals, principal ideals, the ideal class group and its finiteness.
Unique factorisation of ideals.
Explicit factorisation of rational primes in rings of integers of number fields.
Minkowski's constant, calculating the ideal class group. Quadratic examples and applications to diophantine equations.
Units. The logarithmic embedding, statement of Dirichlet's unit theorem.
Formula for counting ideals of bounded norm, rough idea of proof. Dedekind zeta function and Dirichlet's class number formula.