In
mathematics, the (field) norm is a particular mapping defined in
field theory, which maps elements of a larger field into a subfield.
Formal definition
Let ''K'' be a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and ''L'' a finite
extension (and hence an
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
) of ''K''.
The field ''L'' is then a finite dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over ''K''.
Multiplication by α, an element of ''L'',
:
:
,
is a ''K''-
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
of this
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
into itself.
The norm, N
''L''/''K''(''α''), is defined as the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of this
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
.
If ''L''/''K'' is a
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
, one may compute the norm of α ∈ ''L'' as the product of all the
Galois conjugates of α:
:
where Gal(''L''/''K'') denotes the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of ''L''/''K''. (Note that there may be a repetition in the terms of the product.)
For a general
field extension ''L''/''K'', and nonzero α in ''L'', let ''σ''(''α''), ..., σ(''α'') be the roots of the
minimal polynomial of α over ''K'' (roots listed with multiplicity and lying in some extension field of ''L''); then
:
.
If ''L''/''K'' is
separable, then each root appears only once in the product (though the exponent, the
degree 'L'':''K''(α) may still be greater than 1).
Examples
Quadratic field extensions
One of the basic examples of norms comes from
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
extensions
where
is a square-free integer.
Then, the multiplication map by
on an element
is
:
The element
can be represented by the vector
:
since there is a direct sum decomposition
as a
-vector space.
The
matrix of
is then
:
and the norm is
, since it is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of this
matrix.
Norm of Q(√2)
Consider the
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
.
The
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of
over
has order
and is generated by the element which sends
to
. So the norm of
is:
:
The field norm can also be obtained without the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
.
Fix a
-basis of
, say:
:
.
Then multiplication by the number
sends
:1 to
and
:
to
.
So the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of "multiplying by
" is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the
matrix which sends the vector
:
(corresponding to the first basis element, i.e., 1) to
,
:
(corresponding to the second basis element, i.e.,
) to
,
viz.:
:
The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of this
matrix is −1.
''p''-th root field extensions
Another easy class of examples comes from
field extensions of the form
where the prime factorization of
contains no
-th powers, for
a fixed odd prime.
The multiplication map by