Field norm
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In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.
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[edit] Formal definition
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, which are different pairwise, since the extension is Galois and so the minimal polynomial is separate. 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.
[edit] Example
The field norm from the complex numbers to the real numbers sends
- x + iy
to
- x2 + y2.
[edit] 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 minimal 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 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.
