Algebraic Number Theory – Lecture 16
Lee Butler
“It is impossible to travel faster than the speed of light, and certainly not
desirable, as one’s hat keeps blowing off.”
– Woody Allen
1. Numbers, units, and ideals
Algebraic number theory is first and foremost the study of algebraic numbers using algebra,
analysis, and other, funkier stuff.
Definition. An (algebraic)number field Kis a field extension K⊃Qsuch that the degree [K:Q]
is finite. Elements of a number field are called algebraic numbers.
A number α∈Cis algebraic if and only if there is an irreducible, monic polynomial 0 6=f∈Q[x]
such that f(α) = 0. This fis unique and called the minimal polynomial of α. If f∈Z[x] then α
is called an algebraic integer.
The algebraic integers form a ring O. Given a number field Kthe algebraic integers in Kalso
form a ring, the ring of integers of K,OK.
Definition. The units in OK,O×
K, are elements u∈ OKsuch that uv = 1 for some v∈ OK.
In Zwe can factorise numbers uniquely into primes, in the ring of integers of number fields this
is not the case in general. But it is the case for ideals.
Definition. An ideal aof OKis an abelian subgroup under addition such that aOK⊂a. An ideal
pis called:
•prime if whenever ab ⊂p, either a⊂por b⊂p;
•maximal if p6=OKand p⊂p0⊂ OKimplies p0=por p0=OK.
Maximal ideals are always prime, and in special integral domains called Dedekind domains,
every prime ideal is maximal. OKis always a Dedekind domain so we can factorise ideals in OK
into primes.
Definition. An OK-submodule a⊂Kis called a fractional ideal if there is c∈ OKwith c6= 0
and such that cais an ideal in OK.
Fractional ideals form an abelian multiplicative group, JKsay, while the principal ideals form
a normal subgroup, PK, of JK. Their quotient ClK=JK/PKis called the class group. It is
finite and its size h=|ClK|is called the class number. It measures the expansion in passing
from numbers to ideals, while the group of units measures the contraction in this process. This is
captured by the exact sequence
1→ O×
K→K×→JK→ClK→1.
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