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Norm map

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The mapping of a field into a field , where is a finite extension of (cf. Extension of a field), that sends an element to the element that is the determinant of the matrix of the -linear mapping that takes to . The element is called the norm of the element .

One has if and only if . For any ,

that is, induces a homomorphism of the multiplicative groups , which is also called the norm map. For any ,

The group is called the norm subgroup of , or the group of norms (from into ). If is the characteristic polynomial of relative to , then

Suppose that is separable (cf. Separable extension). Then for any ,

where the are all the isomorphisms of into the algebraic closure of fixing the elements of $k$.

The norm map is transitive. If and are finite extensions, then

for any .

References

[1] S. Lang, "Algebra" , Addison-Wesley (1984)
[2] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)
How to Cite This Entry:
Norm map. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Norm_map&oldid=24339
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article