** Next:** Diophantine equations
** Up:** bamc
** Previous:** Basic facts

It's easier to study algebraic numbers as part of a larger structure than
on their own. So we define a *number field* to be the smallest set
containing
plus some finite set
of algebraic
numbers, which is also closed under addition, subtraction, multiplication
and division.
We'll usually denote this number field
.
We define a *ring of integers* to be the set of algebraic
integers in a number field.
WARNING: it's not always obvious what the ring of integers in a number
field is. Take the example
( positive or negative).
If
, then
is an algebraic integer; more generally, the integers
in the number field are
for rational integers of the
same parity. If
, then the only integers in
the number field are the obvious ones
for
rational integers.
One nice property about the rational integers is unique factorization.
Is unique factorization true for other rings of integers? Sometimes yes,
sometimes no.
To make that precise, we'll need some more definitions.
define a *unit* to be an algebraic integer whose reciprocal is
also an algebraic integer. (For example, roots of unity are units, but there
are other units too; we'll see some later.)
We call an element of a ring of integers *irreducible* if
whenever you write
as the product of two elements
of the ring, one of or is a unit. (I didn't say ``prime''
because I'm saving that word for later).
We say a ring of integers has *unique factorization* if whenever
an element of a ring of integers is expressed as a product of irreducible
elements, that expression is unique up to changing the order and multiplying
by units.
For example, the Gaussian integers have unique factorization,
because they admit an analogue of the Euclidean division algorithm.

**Theorem 2**
Given Gaussian integers

and

with

,
there exist Gaussian integers

and

with

and

.

*Proof*.
Draw the square with vertices

. Then

is congruent to a Gaussian integer

inside (or on the boundary of) the square.
Also, the open discs of radius

centered at

cover
the square completely, so

is within

of one corner of the square,
say

. Now take

; then

and

,
so we can set

and we're done.

**Corollary 1**
Every (rational) prime congruent to 1 modulo 4 is the sum of two squares;
moreover, this expression is unique up to order and signs.

*Proof*.
If

, then there exist

and

such that

is divisible by

but not by

. Apply the Euclidean algorithm
in the Gaussian integers (left for you to write down!) to

and

;
the result will be a Gaussian integer

with

. Uniqueness
is also left to you.

EXERCISE: Find some other rings of integers which have unique factorization.
(For starters, try
and
.
On the other hand, consider this example in
:
None of 2, 3, or
can be written as a nontrivial
product of two elements of
, so this ring doesn't have
unique factorization. What to do?
Kummer realized that one could find an algebraic integer in a bigger ring
that would allow you to break up such problem factorizations. (For example,
if we toss in , then it divides both
and .)
However, it turns out that it's a little better to work not
with these ``ideal numbers'', as Kummer called them, but with the collection
of their multiples.
Definition: an *ideal* in a ring of integers is a subset
such that
- for , ;
- if and , then .

Example: If
, then an ideal is an arithmetic progression containing 0.
More general example: the *principal ideal* generated by consists
of all multiples of . But not all ideals have this form!
Because of the way an ideal is defined, we can work ``modulo'' an ideal, that is,
it makes sense to write
because this equivalence respects
addition and multiplication. If is nonzero, then the number of equivalence
classes modulo is finite; we call this number the *norm* of the ideal.
An ideal is *prime* if implies or . For
example, if
and , then is prime if and only if is prime. In general, if has prime norm, it is a prime ideal, but the converse
is not true; we only know that has prime power norm. For example, the
ideal in
is prime, but its norm is 9.
The arithmetic on
is not the same as on
, though!
The main distinction is that in
,
everything not congruent to 0 mod
has a multiplicative inverse. (Thus
is an example of a
*finite field*.)
The big theorem about prime ideals is the recovery of unique
factorization.

**Theorem 3**
Every nonzero ideal in a ring of integers has a unique prime factorization.

**Corollary 2**
If every ideal of a ring of integers

is principal, then

has unique
factorization. (Note: the converse is also true.)

Two ideals and in the ring of integers of a number field
are *equivalent*
if there exists such
that . (Note that need not lie in . If you prefer a definition
within : and are equivalent if there exist nonzero
such that .)
This equivalence is respected by multiplication.

**Theorem 4** (Minkowski)
The number of equivalence classes of ideals in a ring of integers is finite.

This number is called the *class number* of the number field.

**Theorem 5** (Gauss)
For

not divisible by 4,
the number of primitive (having no common factor)
triples

of integers such that

is equal to
12 times the class number of

if

,
or 24 times the class number of

if

.

Note: Gauss didn't express this theorem in terms of number fields,
but in terms of binary quadratic forms
whose discriminant
equals , if
,
or , if
. Two forms are equivalent if you can get
from one to the other by making a variable substitution of the form
where are integers with .
EXERCISE: Prove that
the number of equivalence classes of forms equals the class number
of
.

** Next:** Diophantine equations
** Up:** bamc
** Previous:** Basic facts
Zvezdelina Stankova-Frenkel
2001-01-14