Field norm

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
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, N"L"/"K"("α"), is defined as the determinant of this linear transformation.
If "L"/"K" is a Galois extension, one may compute the norm of α ∈ "L" as the product of all the Galois conjugates of α:
where Gal("L"/"K") denotes the Galois group of "L"/"K". (Note that there may be a repetition in the terms of the product)
For a general field extension "L"/"K", and nonzero α in "L",
let "σ"("α"), ..., σ("α") be the roots of the minimal polynomial of α over "K" (roots listed with multiplicity and lying in some extension field of "L"); then
If "L"/"K" is separable, then each root appears only once in the product (though the exponent, the degree ["L":"K"(α)], may still be greater than 1).
Examples.
Quadratic field extensions.
One of the basic examples of norms comes from quadratic field extensions formula_5 where formula_6 is a square-free integer.
Then, the multiplication map by formula_7 on an element formula_8 is
The element formula_8 can be represented by the vector
since there is a direct sum decomposition formula_12 as a formula_13-vector space.
The matrix of formula_14 is then
and the norm is formula_16, since it is the determinant of this matrix.
Norm of Q(√2).
In this example the norm was the square of the usual Euclidean distance norm in formula_17.
In general, the field norm is very different from the usual distance norm.
We will illustrate that with an example where the field norm can be negative.
Consider the number field formula_18.
The Galois group of formula_19 over formula_13 has order formula_21 and is generated by the element which sends formula_22 to formula_23.
So the norm of formula_24 is:
The field norm can also be obtained without the Galois group.
Fix a formula_13-basis of formula_27, say:
Then multiplication by the number formula_24 sends
So the determinant of "multiplying by formula_24" is the determinant of the matrix which sends the vector
viz.:
The determinant of this matrix is −1.
"K"-th root field extensions.
Another easy class of examples comes from field extensions of the form formula_40 where the prime factorization of formula_41 contains no formula_42-th powers.
The multiplication map by formula_43 of an element isformula_44giving the matrixformula_45The determinant gives the norm
Complex numbers over the reals.
The field norm from the complex numbers to the real numbers sends
to
because the Galois group of formula_47 over formula_48 has two elements,
and taking the product yields .
Finite fields.
Let "L" = GF("q""n") 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,
Properties of the norm.
Several properties of the norm function hold for any finite extension.
Group homomorphism.
The norm N : "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 formula_54
Composition with field extensions.
Additionally, the 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.
Reduction of the norm.
The norm of an element in an arbitrary field extension can be reduced to an easier computation if the degree of the field extension is already known. This isformula_56For example, for formula_57 in the field extension formula_58, the norm of formula_59 isformula_60since the degree of the field extension formula_61 is formula_62.
Detection of units.
An element formula_63 is a unit if and only if formula_64.
For instance
where
Then any number field formula_67 containing formula_68 has it as a unit.
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 a nonzero ideal of "O""K", the ring of integers of the number field "K", N("I") is the number of residue classes in formula_69 – i.e. the cardinality of this finite ring.
Hence this ideal norm is always a positive integer.
When "I" is a principal ideal "αOK" then N("I") is equal to the absolute value of the norm to "Q" of "α", for "α" an algebraic integer.