Frobenius map

From Citizendium
Revision as of 23:17, 16 February 2009 by Todd Coles (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In algebra, the Frobenius map is the p-th power map considered as acting on commutative algebras or fields of prime characteristic p.

We write and note that in characterstic p we have so that F is a ring homomorphism. A homomorphism of fields is necessarily injective, since it is a ring homomorphism with trivial kernel, and a field, viewed as a ring, has no non-trivial ideals. An endomorphism of a field need not be surjective, however. An example is the Frobenius map applied to the rational function field , which has as image the proper subfield .

Frobenius automorphism

When F is surjective as well as injective, it is called the Frobenius automorphism. One important instance is when the domain is a finite field.