Computing Explicit Isomorphisms with Full Matrix Algebras over $$mathbb {F}_q(x)$$ F q ( x )

We propose a polynomial time f-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $$mathbb {f}_q$$ ) for computing an isomorphism (if there is any) of a finite-dimensional $$mathbb {f}_q(x)$$ -algebra $$mathcal{a}$$ given by structure constants with the algebra of n by n matrices with entries from $$mathbb {f}_q(x)$$ . the method is based on computing a finite $$mathbb {f}_q$$ -subalgebra of $$mathcal{a}$$ which is the intersection of a maximal $$mathbb {f}_q[x]$$ -order and a maximal r-order, where r is the subring of $$mathbb {f}_q(x)$$ consisting of fractions of polynomials with denominator having degree not less than that of the numerator.