Frobenius map: Difference between revisions
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In [[algebra]], the '''Frobenius map''' is the ''p''-th power map considered as acting on algebras or fields of [[prime number|prime]] [[characteristic of a field|characteristic]] ''p''. | In [[algebra]], the '''Frobenius map''' is the ''p''-th power map considered as acting on [[commutativity|commutative]] algebras or fields of [[prime number|prime]] [[characteristic of a field|characteristic]] ''p''. | ||
We write <math>F:x \mapsto x^p</math> and note that in characterstic ''p'' we have <math>(x+y)^p = x^p + y^p</math> so that ''F'' is a [[ring homomorphism]]. A homomorphism of fields is necessarily [[injective function|injective]], since it is a [[ring homomorphism]] with trivial kernel, and a field, viewed as a [[ring theory|ring]], has no non-trivial [[ideal]]s. An [[endomorphism]] of a field need not be [[surjective function|surjective]], however. An example is the Frobenius map applied to the [[rational function]] field <math>\mathbf{F}_p(X)</math>, which has as image the proper subfield <math>\mathbf{F}_p(X^p)</math>. | We write <math>F:x \mapsto x^p</math> and note that in characterstic ''p'' we have <math>(x+y)^p = x^p + y^p</math> so that ''F'' is a [[ring homomorphism]]. A homomorphism of fields is necessarily [[injective function|injective]], since it is a [[ring homomorphism]] with trivial kernel, and a field, viewed as a [[ring theory|ring]], has no non-trivial [[ideal]]s. An [[endomorphism]] of a field need not be [[surjective function|surjective]], however. An example is the Frobenius map applied to the [[rational function]] field <math>\mathbf{F}_p(X)</math>, which has as image the proper subfield <math>\mathbf{F}_p(X^p)</math>. | ||
Revision as of 08:34, 22 December 2008
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.