See Frobenius endomorphism on Wiktionary
{
"etymology_text": "Named after German mathematician Ferdinand Georg Frobenius.",
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"ref": "2003, Claudia Miller, “The Frobenius endomorphism and homological dimensions”, in Luchezar L. Avramov, Marc Chardin, Marcel Morales, Claudia Polini, editors, Commutative Algebra: Interactions with Algebraic Geometry: International Conference, American Mathematical Society, page 208:",
"text": "Section 3 concerns what properties of the ring other than regularity are reflected by the homological properties of the Frobenius endomorphism.",
"type": "quote"
},
{
"text": "2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11,\nLet k=◌̅ 𝔽_𝕡, and let q be a power of p such that the group G is defined over 𝔽_𝕢. We then denote by F:G→G the corresponding Frobenius endomorphism. The Lie algebra 𝒢 and the adjoint action of G on 𝒢 are also defined over 𝔽_𝕢 and we still denote by F:𝒢→𝒢 the Frobenius endomorphism on 𝒢.\n[…] Assume that H,X and the action of H over X are all defined over 𝔽_𝕢. Let F:X→X and F:H→H be the corresponding Frobenius endomorphisms."
},
{
"text": "2006, Christophe Doche, Tanja Lange, Chapter 15: Arithmetic of Special Curves, Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren (editors), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Taylor & Francis (Chapman & Hall / CRC Press), page 356,\nThe first attempt to use the Frobenius endomorphism to compute scalar multiples was made by Menezes and Vanstone (MEVA 1900) using the curve\nE:y²+y=x³.\nIn this case, the characteristic polynomial of the Frobenius endomorphism denoted by ϕ₂ (cf. Example 4.87 and Section 13.1.8), which sends P_∞ to itself and (x_1,y_1) to (x,y), is\nχ_E(T)=T²+2.\nThus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points P∈E( 𝔽_2ᵈ), we have ϕ=-[2]P."
}
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"Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R."
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"qualifier": "commutative algebra; field theory; commutative algebra; field theory",
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"(algebra, commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R."
],
"related": [
{
"word": "Frobenius automorphism"
},
{
"word": "Frobenius closure"
},
{
"word": "Frobenius element"
},
{
"word": "Frobenius morphism"
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"synonyms": [
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"word": "Frobenius homomorphism"
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"code": "fr",
"lang": "French",
"sense": "particular endomorphism on a commutative ring with prime characteristic",
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"word": "endomorphisme de Frobenius"
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"code": "de",
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"sense": "particular endomorphism on a commutative ring with prime characteristic",
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"word": "Frobeniushomomorphismus"
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"code": "it",
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"word": "Frobenius endomorphism"
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{
"etymology_text": "Named after German mathematician Ferdinand Georg Frobenius.",
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"text": "Section 3 concerns what properties of the ring other than regularity are reflected by the homological properties of the Frobenius endomorphism.",
"type": "quote"
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{
"text": "2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11,\nLet k=◌̅ 𝔽_𝕡, and let q be a power of p such that the group G is defined over 𝔽_𝕢. We then denote by F:G→G the corresponding Frobenius endomorphism. The Lie algebra 𝒢 and the adjoint action of G on 𝒢 are also defined over 𝔽_𝕢 and we still denote by F:𝒢→𝒢 the Frobenius endomorphism on 𝒢.\n[…] Assume that H,X and the action of H over X are all defined over 𝔽_𝕢. Let F:X→X and F:H→H be the corresponding Frobenius endomorphisms."
},
{
"text": "2006, Christophe Doche, Tanja Lange, Chapter 15: Arithmetic of Special Curves, Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren (editors), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Taylor & Francis (Chapman & Hall / CRC Press), page 356,\nThe first attempt to use the Frobenius endomorphism to compute scalar multiples was made by Menezes and Vanstone (MEVA 1900) using the curve\nE:y²+y=x³.\nIn this case, the characteristic polynomial of the Frobenius endomorphism denoted by ϕ₂ (cf. Example 4.87 and Section 13.1.8), which sends P_∞ to itself and (x_1,y_1) to (x,y), is\nχ_E(T)=T²+2.\nThus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points P∈E( 𝔽_2ᵈ), we have ϕ=-[2]P."
}
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"Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R."
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"(algebra, commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R."
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"code": "fr",
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"sense": "particular endomorphism on a commutative ring with prime characteristic",
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"word": "endomorphisme de Frobenius"
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Download raw JSONL data for Frobenius endomorphism meaning in All languages combined (4.0kB)
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