A mapping
from
to the set
of real numbers, which satisfies the following conditions:
1)
,
and
if and only if
;
2)
;
3)
.
Hence
;
.
The norm of
is often denoted by
instead of
.
A norm is also called an
absolute value
or a
multiplicative valuation.
Norms may (more generally) be considered on any
ring with values in a linearly ordered ring
[4].
See also
Valuation.
Examples of norms.
If
,
the field of real numbers, then
,
the ordinary
absolute value
or
modulus
of the number
,
is a norm. Similarly, if
is the field
of complex numbers or the skew-field
of quaternions, then
is a norm. The subfields of these fields are thus
also provided with an induced norm. Any field has the
trivial norm:
Finite fields and their algebraic extensions only have the trivial norm.
Examples of norms of another type are provided by
logarithmic valuations of a field
:
If
is a valuation on
with values in the group
and if
is a real number,
,
then
is a norm. For example, if
and
is the
-adic
valuation of the field
,
then
is called the
-adic absolute value
or the
-adic norm.
These absolute values satisfy the following
condition, which is stronger than 3):
4)
.
Norms satisfying condition 4) are known as
ultra-metric norms
or
non-Archimedean norms
(as distinct from
Archimedean norms
which do not satisfy this condition (but do satisfy
3)). They are distinguished by the fact that
for all integers
.
All norms on a field of characteristic
are ultra-metric. All ultra-metric norms are obtained from valuations as indicated above:
(and conversely,
can always be taken as a valuation).
A norm
defines a
metric
on
if
is taken as the distance between
and
,
and in this way it defines a topology on
.
The topology of any locally compact field is defined by some norm. Two norms
and
are said to be equivalent if they define the same topology; in a such case there exists a
such that
for all
.
The structure of all Archimedean norms is given by
Ostrowski's theorem:
If
is an Archimedean norm on a field
,
then there exists an isomorphism of
into a certain everywhere-dense subfield of one of the fields
,
or
such that
is equivalent to the norm induced by that of
,
or
.
Any non-trivial norm of the field
of rational numbers is equivalent either to a
-adic
norm
,
where
is a prime number, or to the ordinary norm. For any rational number
one has
A similar formula is also valid for algebraic number fields
[2],
[3].
If
is a norm on a field
,
then
may be imbedded by the classical completion process in a field
that is complete with respect to the norm that (uniquely) extends
(cf.
Complete topological space).
One of the principal modern methods in the study of fields is the imbedding of a field
into the direct product
of all completions
of the field
with respect to all non-trivial norms of
(see
Adèle).
If
admits non-trivial valuations, then it is dense in
in the adèlic topology; in fact, if
are non-trivial, non-equivalent norms on
,
if
are elements of
and if
,
then there exists an
such that
for all
(the
approximation theorem for norms).
A norm on a field
may be extended (in general, non-uniquely) to any algebraic field extension of the field
.
If
is complete with respect to the norm
and if
is an extension of
of degree
,
the extension of
to
is unique, and is given by the formula
for
.