Let K be an algebraic number field of degree d and discriminant ∆ over Q. Let A be an associative algebra over K given by structure constants such that A ∼ = M n (K) holds for some positive integer n. Suppose that d, n and |∆| are bounded. Then an isomorphism A → 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 ismorphisms of central simple algebras of bounded degree over K.Theorem 2. Let A be a Q-subalgebra of M n (R) isomorphic to M n (Q) and let Λ be a maximal Z-order in A. Then there exists an element C ∈ Λ which has rank 1 as a matrix, and whose Frobenius norm C is less than n.Remark 3. When we apply the above theorem, the Frobenius norm · will be inherited from M n (R), with respect to an arbitrary embedding of A into M n (R). Recall that for a matrix X ∈ M n (R) we have X = T r(X T X).Proof. The isomorphism A ∼ = M n (Q) extends to an automorphism of M n (R). Therefore, by the Noether-Skolem Theorem, there exists a matrix P ∈ M n (R) such that A = P M n (Q)P −1 . Let Λ ′ denote the standard maximal order M n (Z) in M n (Q). The theory of maximal orders in central
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