Splitting full matrix algebras over algebraic number fields

Abstract

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

Splitting quaternion algebras over quadratic number fields

“…From this data a zero divisor in A can be constructed via the description in Proposition 7. The algorithms from [10], [20] and [13] generalize these results to matrix algebras of higher degree.…”

“…Although the problem comes from computational representation theory, it has various applications in computational algebraic geometry and number theory as well. The case where K = Q, has connections with explicit n-descent on elliptic curves [5], solving norm equations [13] and parametrizing Severi-Brauer surfaces [10]. In [11] we consider the case where K = F q (t) which is connected to the factorization problem in a certain skew-polynomial ring [8], [9].…”

“…Ivanyos, Rónyai and Schicho proposed an ff-algorithm for the case where K is an algebraic number field [13]. An ff-algorithm is allowed to call an oracle for factoring an integer or polynomial over a finite field at a cost of the size of the input to the oracle call.…”

“…It is natural to consider ff-algorithms for this task as Rónyai showed that the problem of computing an explicit isomorphism between A and M 2 (Q) is at least as hard as factoring integers [21]. The algorithm from [13] however depends exponentially on the degree of the number field, the dimension of the matrix algebra and the logarithm of the discriminant of the number field. This algorithm was improved in [12].…”

Algorithms based on *-algebras, and their applications to isomorphism of polynomials with one secret, group isomorphism, and polynomial identity testing

“…Actually, there is a randomized polynomial time reduction from testing whether a simple algebra over a number field F is isomorphic with a full matrix algebra over F to factoring integers, see [Rón92] and [IR93]. However, for the constructive version of isomorphisms with full matrix algebras such a reduction is only known for the case M (n, K) where n is bounded by a constant, and K is from a finite collection of number fields [IRS12]. Therefore, to determine the relation between the complexity of the isometry problem and that of factoring, it might be useful to devise an alternative approach which gets around constructing explicit isomorphisms with full matrix algebras.…”