Splitting full matrix algebras over algebraic number fields

TL;DR

This paper presents a polynomial-time algorithm for constructing isomorphisms between full matrix algebras over algebraic number fields, given bounded parameters, advancing computational algebra methods.

Contribution

It introduces a polynomial-time ff-algorithm for isomorphism construction of matrix algebras over algebraic number fields with bounded degree and discriminant.

Findings

Polynomial-time ff-algorithm for algebra isomorphism construction

Algorithm applies to central simple algebras of bounded degree

Efficient computation over algebraic number fields

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.