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 and L a finite extension (and hence an algebraic extension) of K.

The field L is then a finite dimensional vector space over K.

Multiplication by α, an element of L,

,
is a K-linear transformation of this vector space into itself.

The norm, NL/K(α), is defined as the determinant of this linear transformation.

If L/K is a Galois extension, 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 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 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 of this matrix.

Norm of Q(√2) 

Consider the number field .

The Galois group 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.

Fix a -basis of , say:
.

Then multiplication by the number  sends
1 to  and
 to .

So the determinant of "multiplying by " is the determinant 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 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  of an element isgiving the matrixThe determinant gives the norm

Complex numbers over the reals 
The field norm from the complex numbers to the real numbers sends

 

to

 ,

because the Galois group of  over  has two elements,
 the identity element and
 complex conjugation,
and taking the product yields .

Finite fields 
Let L = GF(qn) be a finite extension of a finite field K = GF(q). 

Since L/K is a Galois extension, if α is in L, then the norm of α is the product of all the Galois conjugates of α, i.e.

 

In this setting we have the additional properties,

Properties of the norm
Several properties of the norm function hold for any finite extension.

Group homomorphism 
The norm N : L* → K* is a group homomorphism from the multiplicative group of L to the multiplicative group of K, that is

Furthermore, if a in K:

If a ∈ K then

Composition with field extensions 
Additionally, the norm behaves well in towers of fields:

if M is a finite extension of L, then the norm from M to K is just the composition of the norm from M to L with the norm from L to K, i.e.

Reduction of the norm 
The norm of an element in an arbitrary field extension can be reduced to an easier computation if the degree of the field extension is already known. This isFor example, for  in the field extension , the norm of  issince the degree of the field extension  is .

Detection of units 
For  the ring of integers of an algebraic number field , an element  is a unit if and only if .

For instance

where
.

Thus, any number field  whose ring of integers  contains  has it as a unit.

Further properties
The norm of an algebraic integer is again an integer, because it is equal (up to sign) to the constant term of the characteristic polynomial.

In algebraic number theory one defines also norms for ideals. This is done in such a way that if I is a nonzero ideal of OK, the ring of integers of the number field K, N(I) is the number of residue classes in  – i.e. the cardinality of this finite ring. Hence this ideal norm is always a positive integer.

When I is a principal ideal αOK then N(I) is equal to the absolute value of the norm to Q of α, for α an algebraic integer.

See also
 Field trace
 Ideal norm
 Norm form

Notes

References
 
 
 
 

Algebraic number theory