In this section we develop some basic properties of norms, traces, and
discriminants, and give more properties of rings of integers in the
general context of Dedekind domains.
Before discussing norms and traces we introduce some notation for
field extensions. If
are number fields, we let
denote the dimension of viewed as a -vector space. If is a
number field and
, let be the extension of
generated by , which is the smallest number field that contains
both and . If
then has a minimal polynomial
, and the *Galois conjugates* of are the roots
of . The are called the Galois conjugates because the are the orbit
of under the action of
.

*Example 2.4.1*
The element

has minimal polynomial

and the Galois
conjugates are

and

. The cube root

has minimial polynomial

and three Galois conjugates

, where

is a cube root of unity.

We create the extension
in *SAGE*.

sage: L.<cuberoot2> = CyclotomicField(3).extension(x^3 - 2)
sage: cuberoot2^3
2

Then we list the Galois conjugates of
.

sage: cuberoot2.galois_conjugates()
[cuberoot2, (-zeta3 - 1)*cuberoot2, zeta3*cuberoot2]

Note that
:

sage: zeta3 = L.base_field().0
sage: zeta3^2
-zeta3 - 1

Suppose
is an inclusion of number fields and let . Then left multiplication by defines a -linear
transformation
. (The transformation is
-linear because is commutative.)

**Definition 2.4.2** (Norm and Trace)
The

*norm* and

*trace* of

from

to

are

and

We know from linear algebra that
determinants are multiplicative
and traces are additive, so for we have

and

Note that if
is the characteristic polynomial of ,
then the constant term of is
, and the
coefficient of
is
.

*Proof*.
We prove the proposition by computing the characteristic
polynomial

of

. Let

be the minimal polynomial
of

over

, and note that

has distinct roots and is
irreducible, since it is the polynomial in

of least degree
that is satisfied by

and

has characteristic

0. Since

is irreducible, we have

, so

.
Also

satisfies a polynomial if and only if

does, so the
characteristic polynomial of

acting on

is

. Let

be a basis for

over

and note that

is a basis for

, where

. Then

is a basis for

over

, and left multiplication by

acts the same way on the span of

as on the span of

, for any pair

. Thus the matrix of

on

is a block direct sum
of copies of the matrix of

acting on

, so the
characteristic polynomial of

on

is

. The
proposition follows because the roots of

are exactly
the images

, with multiplicity

(since each
embedding of

into

extends in exactly

ways
to

).

It is important in Proposition 2.4.3 that
the product and sum be over *all* the images
,
not over just the distinct images. For example, if , then
, whereas the sum of the distinct conjugates
of is .

The following corollary asserts that the norm and trace behave well in
towers.

**Corollary 2.4.4**
*
Suppose
is a tower of number fields, and
let . Then
*

and

*Proof*.
For the first equation, both sides are the product of

,
where

runs through the embeddings of

into

that fix

. To see this, suppose

fixes

.
If

is an extension of

to

, and

are the embeddings of

into

that fix

, then

are exactly the extensions
of

to

. For the second statement, both sides are the
sum of the

.

The norm and trace down to
of an algebraic integer is an
element of
, because the minimal polynomial of has integer
coefficients, and the characteristic polynomial of is a power of the
minimal polynomial, as we saw in the proof of
Proposition 2.4.3.

**Proposition 2.4.5**
*
Let be a number field. The ring of integers is a lattice
in , i.e.,
and is an abelian group of rank
.*
*Proof*.
We saw in Lemma

2.3.15 that

. Thus there exists a
basis

for

, where each

is in

.
Suppose that as

varies over all elements of

the denominators of the coefficients

are arbitrarily
large. Then subtracting off integer multiples of the

, we see
that as

varies over elements of

with

between

0 and

, the denominators of the

are also
arbitrarily large. This implies that there are infinitely many elements
of

in the bounded subset

Thus for any

, there are elements

such that the
coefficients of

are all less than

(otherwise the elements
of

would all be a ``distance''
of least

from each other, so only finitely
many of them would fit in

).

As mentioned above, the norms of elements of are integers.
Since the norm of an element is the determinant of left multiplication
by that element, the norm is a homogenous polynomial of degree in
the indeterminate coefficients , which is 0 only on the
element 0. If the get arbitrarily small for elements of
, then the values of the norm polynomial get arbitrarily small,
which would imply that there are elements of with positive norm
too small to be in
, a contradiction. So the set contains
only finitely many elements of . Thus the denominators of the
are bounded, so for some , we have that has finite
index in
. Since
is isomorphic to
, it follows from the structure theorem for
finitely generated abelian groups that is isomorphic as a
-module to
, as claimed.

*Proof*.
By Proposition

2.4.5, the ring

is
finitely generated as a module over

, so it is certainly
finitely generated as a ring over

. By Theorem

2.2.9,

is noetherian.

William Stein
2012-09-24