# 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. L can be viewed as a vector space over K. Multiplication by α, an element of L,

$m_{\alpha }:L\to L{\text{ given by }}m_{\alpha }(x)=\alpha x$ ,

is a K-linear transformation of this vector space into itself. The norm, NL/K(α), is defined as the determinant of this linear transformation.

For α in L, let σ1(α), ..., σn(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of L), then

$\operatorname {N} _{L/K}(\alpha )=\left(\prod _{j=1}^{n}\sigma _{j}(\alpha )\right)^{[L:K(\alpha )]}$ .

If L/K is separable then each root appears only once and the exponent above is one.

More particularly, if L/K is a Galois extension and α is in L, then the norm of α is the product of all the Galois conjugates of α, i.e.

$\operatorname {N} _{L/K}(\alpha )=\prod _{g\in \operatorname {Gal} (L/K)}g(\alpha )$ ,

where Gal(L/K) denotes the Galois group of L/K.

## Example

The field norm from the complex numbers to the real numbers sends

x + iy

to

x2 + y2,

because the Galois group of $\mathbb {C}$ over $\mathbb {R}$ has two elements, the identity element and complex conjugation, and taking the product yields (x + iy)(x - iy) = x2 + y2.

$(1+{\sqrt {2}})(1-{\sqrt {2}})=-1.$ ${\begin{bmatrix}1&2\\1&1\end{bmatrix}}.$ The determinant of this matrix is -1.

## Properties of the norm

Several properties of the norm function hold for any finite extension.

The norm NL/K : LK is a group homomorphism of the nonzero elements of L to the nonzero elements of K, that is

$\operatorname {N} _{L/K}(\alpha \beta )=\operatorname {N} _{L/K}(\alpha )\operatorname {N} _{L/K}(\beta ){\text{ for all }}\alpha ,\beta \in L$ .

Furthermore, if a in K:

$\operatorname {N} _{L/K}(a\alpha )=a^{[L:K]}\operatorname {N} _{L/K}(\alpha ){\text{ for all }}\alpha \in L$ .

Additionally, 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.

$\operatorname {N} _{M/K}=\operatorname {N} _{L/K}\circ \operatorname {N} _{M/L}$ .

## Finite fields

Let L = GF(qn) 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.

$\operatorname {N} _{L/K}(\alpha )=\alpha \bullet \alpha ^{q}\bullet \cdots \bullet \alpha ^{q^{n-1}}=\alpha ^{(q^{n}-1)/(q-1)}$ .

In this setting we have the additional properties,

## 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 an ideal of OK, 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 αOK then N(I) is equal to the absolute value of the norm to Q of α, for α an algebraic integer.