Let L be a separable quadratic extension of either Q or F q (t). We propose efficient algorithms for finding isomorphisms between quaternion algebras over L. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple L-algebra. In order to obtain efficient algorithms in the characteristic 2 case, we propose an algorithm for finding nontrivial zeros of a regular quadratic form in four variables over F 2 k (t).
Let L be a separable quadratic extension of either $${\mathbb {Q}}$$
Q
or $${\mathbb {F}}_q(t)$$
F
q
(
t
)
. We exhibit efficient algorithms for finding isomorphisms between quaternion algebras over L. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple L-algebra.
We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion division division algebras and zero divisors in M 2 (D), the full matrix algebra over a division algebra, given by structure constants. We also provide an implementation of our results in MAGMA which shows that the algorithms are truly practical.
We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion division algebras and zero divisors in 𝑀 2 (𝐷), the full matrix algebra over a division algebra, given by structure constants. We also provide an implementation of our results in MAGMA which shows that the algorithms are truly practical.
CCS CONCEPTS• Computing methodologies → Number theory algorithms; Equation and inequality solving algorithms.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.