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 NL/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
- NL/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
- NL/K(αβ) = NL/K(α)NL/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
NL/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 OK, the ring of integers of the number field K, N(I) is the number of residue classes in OK/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 αOK 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.
