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

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.