### 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.

Original language | English |
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Pages (from-to) | 211-223 |

Number of pages | 13 |

Journal | Journal of Algebra |

Volume | 354 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 15 2012 |

### Fingerprint

### Keywords

- Central simple algebra
- Lattice basis reduction
- Maximal order
- Minkowski's theorem on convex bodies
- N-Descent on elliptic curves
- Parametrization
- Real and complex embedding
- Severi-Brauer surfaces
- Splitting
- Splitting element

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*354*(1), 211-223. https://doi.org/10.1016/j.jalgebra.2012.01.008

**Splitting full matrix algebras over algebraic number fields.** / Ivanyos, Gábor; Rónyai, L.; Schicho, Josef.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 354, no. 1, pp. 211-223. https://doi.org/10.1016/j.jalgebra.2012.01.008

}

TY - JOUR

T1 - Splitting full matrix algebras over algebraic number fields

AU - Ivanyos, Gábor

AU - Rónyai, L.

AU - Schicho, Josef

PY - 2012/3/15

Y1 - 2012/3/15

N2 - 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.

AB - 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.

KW - Central simple algebra

KW - Lattice basis reduction

KW - Maximal order

KW - Minkowski's theorem on convex bodies

KW - N-Descent on elliptic curves

KW - Parametrization

KW - Real and complex embedding

KW - Severi-Brauer surfaces

KW - Splitting

KW - Splitting element

UR - http://www.scopus.com/inward/record.url?scp=84856386989&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856386989&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2012.01.008

DO - 10.1016/j.jalgebra.2012.01.008

M3 - Article

VL - 354

SP - 211

EP - 223

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -