Splitting full matrix algebras over algebraic number fields

2011/06/30 by Gábor Ivanyos, Ivanyos, Gábor, Lajos Rónyai +3
Computer Science · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA) #Symbolic Computation (cs.SC) #cs.SC #math.NT #math.RA

paper · pdf · doi:10.48550/arxiv.1106.6191

15 pages; Theorem 2 and Lemma 8 corrected

arxiv created 2011/12/21 · arxiv updated 2011/12/22

Abstract

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with M_n(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.

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