### Abstract

Let F be a finite field or an algebraic number field. In previous papers we have shown how to find the basic building blocks (the radical and the simple components) of a finite dimensional algebra over F in polynomial time (deterministically in characteristic zero and Las Vegas in the finite case). A finite-dimensional simple algebra A is a full matrix algebra over some not necessarily commutative extension field G of F. The problem remains to find G and an isomorphism between A and a matrix algebra over G. This, too, can be done in polynomial time for finite F. We indicate in the present paper that the problem for F = Q might be substantially more difficult. We link the problem to hard number theoretic problems such as quadratic residuosity modulo a composite number. We show that assuming the generalized Riemann hypothesis, there exists a randomized polynomial time reduction from quadratic residuosity to determining whether or not a given 4-dimensional algebra over Q has zero divisors. It will follow that finding a pair of zero divisors is at least as hard as factoring squarefree integers.

Original language | English |
---|---|

Pages (from-to) | 494-506 |

Number of pages | 13 |

Journal | Journal of Algorithms |

Volume | 9 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1988 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Optimization
- Computational Mathematics
- Computational Theory and Mathematics

### Cite this

*Journal of Algorithms*,

*9*(4), 494-506. https://doi.org/10.1016/0196-6774(88)90014-4