Fix an algebraic closure
of
. For example,
could
be the subfield of the complex numbers
generated by all roots in
of all polynomials with coefficients in
.
Much of this course is about algebraic integers.

**Definition 5.1.1** (Algebraic Integer)
An element

is an

if it is a
root of some monic polynomial with coefficients in

.

**Definition 5.1.2** (Minimal Polynomial)
The

of

is the monic polynomial

of least positive degree such that

.

The minimal polynomial of divides any polynomial such
that
, for the following reason. If
, use
the division algorithm to write , where
. We have
,
so is a root of . However, is the polynomial of least
positive degree with root , so .

**Lemma 5.1.3**
*
If is an algebraic integer, then the minimal polynomial
of has coefficients in
.*
*Proof*.
Suppose

is the minimal polynomial of

and

is a monic integral polynomial such that

.
As mentioned after the definition of minimal polynomial, we have

, for some

. If

, then some prime

divides the denominator of some coefficient of

. Let

be the
largest power of

that divides some denominator of some
coefficient

, and likewise let

be the largest power of

that divides some denominator of a coefficient of

. Then

, and if we reduce both sides modulo

, then the
left hand side is

0 but the right hand side is a product of two
nonzero polynomials in

, hence nonzero, a contradiction.

**Proposition 5.1.4**
*
An element
is integral if and only if
is
finitely generated as a
-module.*
*Proof*.
Suppose

is integral and let

be the monic minimal polynomial
of

(that

is Lemma

5.1.3). Then

is generated by

, where

is
the degree of

. Conversely, suppose

is such that

is finitely generated, say by elements

. Let

be any integer bigger
than the degree of any

. Then there exist integers

such
that

, hence

satisfies
the monic polynomial

, so

is integral.

The rational number
is not integral. Note that
is not a finitely generated
-module, since is infinite
and .

**Proposition 5.1.5**
*
The set
of all algebraic integers is a ring, i.e., the sum and
product of two algebraic integers is again an algebraic integer.*
*Proof*.
Suppose

, and let

be the degrees of the
minimal polynomials of

, respectively. Then

span

and

span

as

-module. Thus
the elements

for

span

. Since

is a submodule of the
finitely-generated module

, it is finitely
generated, so

is integral. Likewise,

is a submodule of

, so it is also finitely
generated and

is integral.

Recall that a is a subfield of
such
that the degree
is finite.

**Definition 5.1.6** (Ring of Integers)
The

of a number field

is the ring

$x$ is an algebraic integer

The field
of rational numbers is a number field of degree ,
and the ring of integers of
is
. The field
of
Gaussian integers has degree and
. The field
has ring of integers
.
Note that the Golden ratio
satisfies .
According to , the ring of integers of
is
, where
.

**Definition 5.1.7** (Order)
An

in

is any subring

of

such that the
quotient

of abelian groups is finite.
(Note that

must contain

because it is a ring, and for us
every ring has a

.)

As noted above,
is the ring of integers of
. For every
nonzero integer , the subring
of
is an order.
The subring
of
is not an order, because
does not
have finite index in
. Also the subgroup
of
is not an order because it is not a ring.
We will frequently consider orders in practice because they are often
much easier to write down explicitly than . For example, if
and is an algebraic integer, then
is an order in , but frequently
.

**Lemma 5.1.8**
*
Let be the ring of integers of a number field. Then
and
.*
*Proof*.
Suppose

with

in lowest terms and

. The monic minimal polynomial of

is

, so
if

then Lemma

5.1.3 implies that

is
not an algebraic integer, a contradiction.

To prove that
, suppose
, and let
be the minimal monic polynomial of . For any
positive integer , the minimal monic polynomial of is
, i.e., the polynomial obtained from by
multiplying the coefficient of
by , multiplying the
coefficient of
by , multiplying the coefficient of
by , etc. If is the least common multiple of
the denominators of the coefficients of , then the minimal monic
polynomial of has integer coefficients, so is
integral and
. This proves that
.

In the next two sections we will develop some basic properties of
norms and traces, and deduce further properties of rings of integers.

William Stein
2004-05-06