# 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${\ displaystyle \ mathbb {F} _ {q ^ {n}} / \ mathbb {F} _ {q}}$
${\ displaystyle x \ mapsto x ^ {1 + q + q ^ {2} + \ ldots + q ^ {n-1}}}$.