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Some Results Concerning the Explicit Isomorphism Problem over Number Fields

  • Péter KutasEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)

Abstract

We consider two problems. First let u be an element of a quaternion algebra B over \(\mathbb {Q}(\sqrt{d})\) such that u is non-central and \(u^2\in \mathbb {Q}\). We relate the complexity of finding an element \(v'\) such that \(uv'=-v'u\) and \(v'^2\in \mathbb {Q}\) to a fundamental problem studied earlier. For the second problem assume that \(A\cong M_2(\mathbb {Q}(\sqrt{d}))\). We propose a polynomial (randomized) algorithm which finds a non-central element \(l\in A\) such that \(l^2\in \mathbb {Q}\). Our results rely on the connection between solving quadratic forms over \(\mathbb {Q}\) and splitting quaternion algebras over \(\mathbb {Q}\) [4], and Castel’s algorithm [1] which finds a rational solution to a non-degenerate quadratic form over \(\mathbb {Q}\) in 6 dimensions in randomized polynomial time. We use these two results to construct a four dimensional subalgebra over \(\mathbb {Q}\) of A which is a quaternion algebra. We also apply our results to analyze the complexity of constructing involutions.

References

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    Castel, P.: Un algorithme de résolution des équations quadratiques en dimension 5 sans factorisation, Ph.D. thesis, October 2011Google Scholar
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    Ivanyos, G., Rónyai, L., Schicho, J.: Splitting full matrix algebras over algebraic number fields. J. Algebra 354, 211–223 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
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    Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The book of involutions. AMS Colloquium Publications, vol. 44, p. 593 (1998)Google Scholar
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    Rónyai, L.: Simple algebras are difficult. In: Proceedings of the 19th Annual ACM Symposium on the Theory of Computing, New York, pp. 398–408 (1987)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and its ApplicationsCentral European UniversityBudapestHungary

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