Computing Explicit Isomorphisms with Full Matrix Algebras over $$\mathbb {F}_q(x)$$ |
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Authors: | Gábor Ivanyos Péter Kutas Lajos Rónyai |
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Institution: | 1.Institute for Computer Science and Control, Hungarian Acad. Sci.,Budapest,Hungary;2.Department of Mathematics and Its Applications,Central European University,Budapest,Hungary;3.Department of Algebra,Budapest University of Technology and Economics,Budapest,Hungary |
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Abstract: | 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 \(\mathbb {F}_qx]\)-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. |
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