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Section: New Results

Quaternion algebras

Participants : Jean-Paul Cerri, Pierre Lezowski, Aurel Page.

In the article [14] written with J. Chaubert, J.-P. Cerri and P. Lezowski study totally indefinite Euclidean quaternion fields over a number field $K$, that is to say where no infinite place is ramified. Relying on some generalisation of Hasse–Schilling–Maaß Norm Theorem, they prove that the Euclidean property of $K$ implies the Euclidean property of any totally indefinite Euclidean quaternion field over $K$. Conversely, they provide the complete list of norm-Euclidean and totally indefinite quaternion fields over an imaginary quadratic number field. In particular, the article exhibits a totally indefinite and norm-Euclidean quaternion field over a non-Euclidean number field. This provides an answer to a question by Eichler. The proofs are both theoretic and algorithmic. The article has been published in Acta Arithmetica.

Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. In the article [25] gives a an algorithm for indefinite quaternion algebras by reducing the decision problem to that in the underlying number field. It also gives an heuristically subexponential algorithm for finding a generator.