# Norm (body extension)

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 .

## definition

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 .${\ displaystyle a, b \ in L}$ Restricted to the multiplicative groups , the norm is therefore a homomorphism
${\ displaystyle N_ {L / K} \ colon L ^ {\ times} \ to K ^ {\ times}.}$ • 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 :${\ displaystyle M / L}$ ${\ displaystyle N_ {L / K}, N_ {M / L}}$ ${\ displaystyle N_ {M / K}}$ ${\ displaystyle N_ {M / K} (a) = N_ {L / K} (N_ {M / L} (a))}$ for everyone .${\ displaystyle a \ in M}$ • 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:${\ displaystyle a \ in L}$ ${\ displaystyle f \ in K [X]}$ ${\ displaystyle d}$ ${\ displaystyle a_ {0} \ in K}$ ${\ displaystyle f}$ ${\ displaystyle r = [L: K (a)]}$ ${\ displaystyle N_ {L / K} (a) = (- 1) ^ {dr} a_ {0} ^ {r}}$ • 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${\ displaystyle L / K}$ ${\ displaystyle [L: K] = qr}$ ${\ displaystyle r}$ ${\ displaystyle \ sigma}$ ${\ displaystyle \ operatorname {Hom} _ {K} (L, {\ bar {K}})}$ ${\ displaystyle K}$ ${\ displaystyle L}$ ${\ displaystyle {\ bar {K}}}$ ${\ displaystyle K}$ ${\ displaystyle a \ in L}$ ${\ displaystyle N_ {L / K} (a) = \ left (\, \ prod _ {i = 1} ^ {r} \ sigma _ {i} (a) \ right) ^ {q}}$ If in particular is Galois with Galois group , this means ${\ displaystyle L / K}$ ${\ displaystyle \ operatorname {Gal} (L / K)}$ ${\ displaystyle N_ {L / K} (a) = \ prod _ {\ sigma \ in \ operatorname {Gal} (L / K)} \ sigma (a).}$ ## Examples

${\ displaystyle N _ {\ mathbb {C} / \ mathbb {R}} (a + ib) = \ sigma _ {1} (a + ib) \ sigma _ {2} (a + ib) = id (a + ib) {\ overline {(a + ib)}} = (a + ib) (a-ib) = a ^ {2} + b ^ {2}}$ .
• The norm of is the picture${\ displaystyle \ mathbb {Q} ({\ sqrt {2}}) / \ mathbb {Q}}$ ${\ displaystyle a + b {\ sqrt {2}} \ mapsto a ^ {2} -2b ^ {2}}$ for .${\ displaystyle a, b \ in \ mathbb {Q}}$ • The norm of is the picture${\ displaystyle \ mathbb {F} _ {q ^ {n}} / \ mathbb {F} _ {q}}$ ${\ displaystyle x \ mapsto x ^ {1 + q + q ^ {2} + \ ldots + q ^ {n-1}}}$ .