Next: Diophantine equations
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
plus some finite set
numbers, which is also closed under addition, subtraction, multiplication
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).
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
One nice property about the rational integers is unique factorization.
Is unique factorization true for other rings of integers? Sometimes yes,
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
For example, the Gaussian integers have unique factorization,
because they admit an analogue of the Euclidean division algorithm.
Given Gaussian integers
there exist Gaussian integers
Draw the square with vertices
is congruent to a Gaussian integer
inside (or on the boundary of) the square.
Also, the open discs of radius
the square completely, so
of one corner of the square,
. Now take
so we can set
and we're done.
Every (rational) prime congruent to 1 modulo 4 is the sum of two squares;
moreover, this expression is unique up to order and signs.
EXERCISE: Find some other rings of integers which have unique factorization.
(For starters, try
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
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
, then there exist
is divisible by
but not by
. Apply the Euclidean algorithm
in the Gaussian integers (left for you to write down!) to
the result will be a Gaussian integer
is also left to you.
, 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
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
is prime, but its norm is 9.
The arithmetic on
is not the same as on
The main distinction is that in
everything not congruent to 0 mod
has a multiplicative inverse. (Thus
is an example of a
The big theorem about prime ideals is the recovery of unique
- for , ;
- if and , then .
Every nonzero ideal in a ring of integers has a unique prime factorization.
Two ideals and in the ring of integers of a number field
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.
If every ideal of a ring of integers
is principal, then
factorization. (Note: the converse is also true.)
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.
Note: Gauss didn't express this theorem in terms of number fields,
but in terms of binary quadratic forms
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
not divisible by 4,
the number of primitive (having no common factor)
of integers such that
is equal to
12 times the class number of
or 24 times the class number of
Next: Diophantine equations
Previous: Basic facts