Field norm

In the field theory the norm of a field extension is a special extension of the associated figure. It makes each element of the larger body from the smaller body.

This standard term is significantly different from the concept of the norm of a normed vector space, it is therefore sometimes in contrast to the vector norm also called body standard.

Definition

It is a finite field extension. A firm selected element defines a linear map

Its determinant is called the norm of written. It is a member of; is the norm so a mapping

Properties

  • Just for applies.
  • The norm is multiplicative, that is,
  • Is another finite field extension, then one has, the three standard functions and, as referred to in the following as transitivity of the norm, relationship:
  • Is so true.
  • If the minimal polynomial of degree, the absolute term of and, then:
  • Includes a finite field extension, where the number of elements in, the amount of all - homomorphisms of the algebraic degree of was. Then for each element

Examples

  • The norm of complex numbers over the real numbers is any complex number down to their absolute square. It is therefore
  • The norm of the mapping
  • The norm of the mapping
Minimal polynomial (linear algebra) Discriminant Hilbert's Theorem 90
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