Field norm

In mathematics, the (field) norm is a mapping defined in field theory (mathematics), 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 Conjugate element (field theory)

: 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 Ideal (ring theory)s. 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 α O 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.