Algorithmic properties of maximal orders in simple algebras over Q

Research output: Article

15 Citations (Scopus)

Abstract

This paper addresses an algorithmic problem related to associative algebras. We show that the problem of deciding if the index of a given central simple algebra {Mathematical expression} over an algebraic number field is d, where d is a given natural number, belongs to the complexity class N P ∩co N P. As consequences, we obtain that the problem of deciding if {Mathematical expression} is isomorphic to a full matrix algebra over the ground field and the problem of deciding if {Mathematical expression} is a skewfield both belong to N P ∩co N P. These results answer two questions raised in [25]. The algorithms and proofs rely mostly on the theory of maximal orders over number fields, a noncommutative generalization of algebraic number theory. Our results include an extension to the noncommutative case of an algorithm given by Huang for computing the factorization of rational primes in number fields and of a method of Zassenhaus for testing local maximality of orders in number fields.

Original languageEnglish
Pages (from-to)225-243
Number of pages19
JournalComputational Complexity
Volume2
Issue number3
DOIs
Publication statusPublished - szept. 1992

Fingerprint

Maximal Order
Algebra
Number field
Number theory
Factorization
Central Simple Algebra
Algebraic number Field
Algebraic number
Complexity Classes
Associative Algebra
Matrix Algebra
Testing
Natural number
Isomorphic
Computing

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Mathematics
  • Mathematics(all)
  • Computational Theory and Mathematics

Cite this

Algorithmic properties of maximal orders in simple algebras over Q. / Rónyai, L.

In: Computational Complexity, Vol. 2, No. 3, 09.1992, p. 225-243.

Research output: Article

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