Splitting full matrix algebras over algebraic number fields

Let K be a fixed algebraic number field and let A be an associative algebra over K given by structure constants such that A≅Mn(K) holds for some positive integer n. Suppose that n is bounded. Then an isomorphism A→Mn(K) can be constructed by a polynomial time ff-algorithm. An ff-algorithm is a deter...

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Vydané v:	Journal of algebra Ročník 354; číslo 1; s. 211 - 223
Hlavní autori:	Ivanyos, Gábor, Rónyai, Lajos, Schicho, Josef
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elsevier Inc 15.03.2012
Predmet:	Central simple algebra Lattice basis reduction Maximal order Minkowskiʼs theorem on convex bodies n-Descent on elliptic curves Parametrization Real and complex embedding Severi–Brauer surfaces Splitting Splitting element Parametrization Splitting Central simple algebra Splitting element Real and complex embedding Minkowskiʼs theorem on convex bodies Lattice basis reduction n-Descent on elliptic curves Severi–Brauer surfaces Maximal order
ISSN:	0021-8693, 1090-266X
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Abstract	Let K be a fixed algebraic number field and let A be an associative algebra over K given by structure constants such that A≅Mn(K) holds for some positive integer n. Suppose that n is bounded. Then an isomorphism A→Mn(K) can be constructed by a polynomial time ff-algorithm. An ff-algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields. As a consequence, we obtain a polynomial time ff-algorithm to compute isomorphisms of central simple algebras of bounded degree over K.
AbstractList	Let K be a fixed algebraic number field and let A be an associative algebra over K given by structure constants such that A≅Mn(K) holds for some positive integer n. Suppose that n is bounded. Then an isomorphism A→Mn(K) can be constructed by a polynomial time ff-algorithm. An ff-algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields. As a consequence, we obtain a polynomial time ff-algorithm to compute isomorphisms of central simple algebras of bounded degree over K.
Author	Schicho, Josef Rónyai, Lajos Ivanyos, Gábor
Author_xml	– sequence: 1 givenname: Gábor surname: Ivanyos fullname: Ivanyos, Gábor email: Gabor.Ivanyos@sztaki.mta.hu organization: Computer and Automation Research Institute, Hungarian Acad. Sci., Kende u. 13-17, H-1111 Budapest, Hungary – sequence: 2 givenname: Lajos surname: Rónyai fullname: Rónyai, Lajos email: Lajos.Ronyai@sztaki.mta.hu organization: Computer and Automation Research Institute, Hungarian Acad. Sci., Kende u. 13-17, H-1111 Budapest, Hungary – sequence: 3 givenname: Josef surname: Schicho fullname: Schicho, Josef email: josef.schicho@oeaw.ac.at organization: Johann Radon Institute of Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria
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Keywords	Parametrization Splitting Central simple algebra Splitting element Real and complex embedding Minkowskiʼs theorem on convex bodies Lattice basis reduction n-Descent on elliptic curves Severi–Brauer surfaces Maximal order
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Snippet	Let K be a fixed algebraic number field and let A be an associative algebra over K given by structure constants such that A≅Mn(K) holds for some positive...
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SubjectTerms	Central simple algebra Lattice basis reduction Maximal order Minkowskiʼs theorem on convex bodies n-Descent on elliptic curves Parametrization Real and complex embedding Severi–Brauer surfaces Splitting Splitting element
Title	Splitting full matrix algebras over algebraic number fields
URI	https://dx.doi.org/10.1016/j.jalgebra.2012.01.008
Volume	354
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