In body theory , the norm of a body expansion is a special mapping assigned to the expansion. It maps every element of the larger body onto the smaller body.

This norm concept differs significantly from the concept of the norm of a normalized vector space , so it is sometimes called a body norm in contrast to the vector norm .

Let it be a finite expansion of the body . A fixed element defines a - linear mapping${\ displaystyle L / K}$${\ displaystyle a \ in L}$${\ displaystyle K}$

${\ displaystyle L \ to L, \ quad x \ mapsto ax.}$

Its determinant is called the norm of , written . She is an element of ; so the norm is a map
${\ displaystyle a}$${\ displaystyle N_ {L / K} (a)}$${\ displaystyle K}$

${\ displaystyle N_ {L / K} \ colon L \ to K, \ quad a \ mapsto N_ {L / K} (a).}$

properties

Exactly applies to .${\ displaystyle a = 0}$${\ displaystyle N_ {L / K} (a) = 0}$

The norm is multiplicative, i.e. H.

${\ displaystyle N_ {L / K} (down) = N_ {L / K} (a) \ cdot N_ {L / K} (b)}$for everyone .${\ displaystyle a, b \ in L}$

${\ displaystyle N_ {L / K} \ colon L ^ {\ times} \ to K ^ {\ times}.}$

If there is a further finite expansion of the body, then one has the three norm functions and , which are related in the following, called the transitivity of the norm :${\ displaystyle M / L}$${\ displaystyle N_ {L / K}, N_ {M / L}}$${\ displaystyle N_ {M / K}}$

Is , then applies .${\ displaystyle a \ in K}$${\ displaystyle N_ {L / K} (a) = a ^ {[L: K]}}$

If the minimal polynomial is of degree , the absolute term of and , then:${\ displaystyle a \ in L}$${\ displaystyle f \ in K [X]}$${\ displaystyle d}$${\ displaystyle a_ {0} \ in K}$${\ displaystyle f}$${\ displaystyle r = [L: K (a)]}$

Is a finite field extension with , with the number of elements in the set of all -Homomorphismen of the algebraic closure of was. Then applies to every element${\ displaystyle L / K}$${\ displaystyle [L: K] = qr}$${\ displaystyle r}$${\ displaystyle \ sigma}$${\ displaystyle \ operatorname {Hom} _ {K} (L, {\ bar {K}})}$${\ displaystyle K}$${\ displaystyle L}$${\ displaystyle {\ bar {K}}}$${\ displaystyle K}$${\ displaystyle a \ in L}$