Field norm - Definition and Overview

In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one. An example is the mapping from the complex numbers to the real numbers sending

x + iy

to

x² + y².

In general if K is a field and L a Galois extension of K, the norm N_L/K of an element α of L is defined as the product of all the conjugates

g(α)

of α, for g in the Galois group G of L/K. Since

N_L/K(α)

is immediately seen to be invariant under G, it follows that it lies in K. It also follows directly from the definition that

N_L/K(αβ) = N_L/K(α)N_L/K(β)

so that the norm, when considered on non-zero elements, is a group homomorphism from the multiplicative group of L to that of K.

The norm of an algebraic element γ over K can be defined directly as the product N(γ) of the roots of its minimal polynomial. Assuming γ is in L, the elements

g(γ)

are those roots, each repeated a certain number d of times. Here

d = [L: M]

is the degree of L over the subfield M of L that is the splitting field of the minimal polynomial of γ. Therefore the relationship of the norms is

N_L/K(γ) = N(γ)^d.

The norm of an algebraic integer is again an integer.

In algebraic number theory one defines also norms for ideals. This is done in such a way that if I is an ideal of O_K, the ring of integers of the number field K, N(I) is the number of residue classes in O_K/I - i.e. the cardinality of this finite ring. Hence this norm of an ideal is always a positive integer. When I is a principal ideal αO_K there is the expected relation between N(I) and the absolute value of the norm to Q of α, for α an algebraic integer.

See also: field trace.