Norm (body extension)

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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

Its determinant is called the norm of , written . She is an element of ; so the norm is a map


  • Exactly applies to .
  • The norm is multiplicative, i.e. H.
for everyone .
Restricted to the multiplicative groups , the norm is therefore a homomorphism
  • 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 :
for everyone .
  • Is , then applies .
  • If the minimal polynomial is of degree , the absolute term of and , then:
  • 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
If in particular is Galois with Galois group , this means


  • The norm of is the picture
for .
  • The norm of is the picture

See also

Individual evidence

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