In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
Contents
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.
For nonzero α in L, let σ1(α), ..., σn(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of L), then
If L/K is separable then each root appears only once in the product (the exponent [L:K(α)] may still be greater than 1).
More particularly, if L/K is a Galois extension and α is in L, then the norm of α is the product of all the Galois conjugates of α, i.e.
where Gal(L/K) denotes the Galois group of L/K.
Example
The field norm from the complex numbers to the real numbers sends
x + iyto
x2 + y2,because the Galois group of
In this example the norm was the square of the usual Euclidean distance norm in
The field norm can also be obtained without the Galois group. Fix a
The determinant of this matrix is −1.
Properties of the norm
Several properties of the norm function hold for any finite extension.
The norm NL/K : 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
Additionally, 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.
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,
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 an ideal of OK, the ring of integers of the number field K, N(I) is the number of residue classes in