In a ring with prime characteristic p, the map raising elements to the pth power
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In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p, an important class that includes finite fields. The endomorphism maps every element to its p-th power. In certain contexts it is an automorphism, but this is not true in general.
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commutative algebra and field theory, the Frobeniusendomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic...
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For...
In mathematics, the Frobeniusendomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping...
of prime characteristic p, R → R, x → xp is a ring endomorphism called the Frobeniusendomorphism. If R and S are rings, the zero function from R to S...
has characteristic 0, or, when k has characteristic p > 0, the Frobeniusendomorphism x ↦ xp is an automorphism of k. The separable closure of k is algebraically...
under the Frobeniusendomorphism F*. Brion & Kumar (2005) give a detailed discussion of Frobenius splittings. A fundamental property of Frobenius-split projective...
of the trace of the Frobeniusendomorphism on its cohomology groups. There are several generalizations: the Frobeniusendomorphism can be replaced by a...
F[X] (where F is assumed to have prime characteristic p). If the Frobeniusendomorphism x ↦ x p {\displaystyle x\mapsto x^{p}} of F is not surjective, there...
If R is an integral domain of prime characteristic p, then the Frobeniusendomorphism x ↦ xp is injective. The Wikibook Abstract algebra has a page on...
Frobenius endomorphism (also known as Frobenius morphism, Frobenius map) Frobenius determinant theorem Frobenius formula Frobenius group Frobenius complement...
is induced by a nondiscrete valuation of rank 1, such that the Frobeniusendomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded...
an endomorphism whose square is the endomorphism αφ associated to the Frobeniusendomorphism φ of the field F. Roughly speaking, this endomorphism απ...
direct product of an empty collection of rings is the zero ring. The endomorphism ring of the trivial group is the zero ring. The ring of continuous real-valued...
duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama...
characteristic n is a (Z/nZ)-algebra in the same way. Given an R-module M, the endomorphism ring of M, denoted EndR(M) is an R-algebra by defining (r·φ)(x) = r·φ(x)...
analogous to the Wrońskian with differentiation replaced by the Frobeniusendomorphism over a finite field. Alternant matrix Vandermonde matrix Peano published...
mathematics. More generally, endomorphism rings of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the Klein four-group...
{\mathcal {O}}(X):x^{p}=0\}} so it is given by the kernel of the Frobeniusendomorphism. Over characteristic 0 there is a nice classification of unipotent...
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