### 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 language | English |
---|---|

Pages (from-to) | 225-243 |

Number of pages | 19 |

Journal | Computational Complexity |

Volume | 2 |

Issue number | 3 |

DOIs | |

Publication status | Published - szept. 1992 |

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### 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.

Research output: Article

*Computational Complexity*, vol. 2, no. 3, pp. 225-243. https://doi.org/10.1007/BF01272075

}

TY - JOUR

T1 - Algorithmic properties of maximal orders in simple algebras over Q

AU - Rónyai, L.

PY - 1992/9

Y1 - 1992/9

N2 - 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.

AB - 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.

KW - Subject classifications: 68Q40, 11Y40, 68Q25, 11Y16

UR - http://www.scopus.com/inward/record.url?scp=0042936679&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042936679&partnerID=8YFLogxK

U2 - 10.1007/BF01272075

DO - 10.1007/BF01272075

M3 - Article

AN - SCOPUS:0042936679

VL - 2

SP - 225

EP - 243

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 3

ER -