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.

${\displaystyle N_{L/K}(ab)=N_{L/K}(a)\cdot N_{L/K}(b)}$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

• The norm of the complex numbers over the real numbers maps each complex number to its absolute square . So it is

${\displaystyle N_{\mathbb {C} /\mathbb {R} }(a+ib)=\sigma _{1}(a+ib)\sigma _{2}(a+ib)=id(a+ib){\overline {(a+ib)}}=(a+ib)(a-ib)=a^{2}+b^{2}}$.

• 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 ${\displaystyle \mathbb {F} _{q^{n}}/\mathbb {F} _{q}}$

${\displaystyle x\mapsto x^{1+q+q^{2}+\ldots +q^{n-1}}}$.

See also

• Track (body extension)
• Discriminant
• Pythagorean body
• Hilbert's Theorem 90

Individual evidence

1. ^ Bosch, Algebra 5th edition, 2004, p. 196ff