Splitting full matrix algebras over algebraic number fields
Copyright policy )Splitting full matrix algebras over algebraic number fields
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.
15 pages; Theorem 2 and Lemma 8 corrected
Computer Science - Symbolic Computation, FOS: Computer and information sciences, real embeddings, Other algebras and orders, and their zeta and \(L\)-functions, Symbolic Computation (cs.SC), Endomorphism rings; matrix rings, Symbolic computation and algebraic computation, Finite-dimensional division rings, Minkowski theorem on convex bodies, parametrizations, n-Descent on elliptic curves, central simple algebras, Minkowskiʼs theorem on convex bodies, Maximal order, Severi-Brauer surfaces, FOS: Mathematics, Number Theory (math.NT), Number-theoretic algorithms; complexity, Central simple algebra, Algebra and Number Theory, Mathematics - Number Theory, Parametrization, Real and complex embedding, Lattice basis reduction, splittings, maximal orders, Severi–Brauer surfaces, Mathematics - Rings and Algebras, Computational aspects of associative rings (general theory), splitting elements, Splitting, complex embeddings, lattice bases reductions, polynomial time ff-algorithms, Rings and Algebras (math.RA), Splitting element, descent on elliptic curves
Computer Science - Symbolic Computation, FOS: Computer and information sciences, real embeddings, Other algebras and orders, and their zeta and \(L\)-functions, Symbolic Computation (cs.SC), Endomorphism rings; matrix rings, Symbolic computation and algebraic computation, Finite-dimensional division rings, Minkowski theorem on convex bodies, parametrizations, n-Descent on elliptic curves, central simple algebras, Minkowskiʼs theorem on convex bodies, Maximal order, Severi-Brauer surfaces, FOS: Mathematics, Number Theory (math.NT), Number-theoretic algorithms; complexity, Central simple algebra, Algebra and Number Theory, Mathematics - Number Theory, Parametrization, Real and complex embedding, Lattice basis reduction, splittings, maximal orders, Severi–Brauer surfaces, Mathematics - Rings and Algebras, Computational aspects of associative rings (general theory), splitting elements, Splitting, complex embeddings, lattice bases reductions, polynomial time ff-algorithms, Rings and Algebras (math.RA), Splitting element, descent on elliptic curves
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