2018 Springer International Publishing AG,
part of Springer Nature
Archiv der Mathematik
Transfer of quadratic forms and of quaternion algebras over
quadratic ﬁeld extensions
Karim Johannes Becher, Nicolas Grenier-Boley,
and Jean-Pierre Tignol
Abstract. Two diﬀerent proofs are given showing that a quaternion al-
gebra Q deﬁned over a quadratic ´etale extension K of a given ﬁeld has
a corestriction that is not a division algebra if and only if Q contains a
quadratic algebra that is linearly disjoint from K. This is known in the
case of a quadratic ﬁeld extension in characteristic diﬀerent from two. In
the case where K is split, the statement recovers a well-known result on
biquaternion algebras due to Albert and Draxl.
Mathematics Subject Classiﬁcation. 11E04, 11E81, 12G05, 16H05.
Keywords. Isotropy, Witt index, Corestriction, Albert form, Character-
1. Introduction. A well-known theorem of Albert states that if a tensor prod-
uct of two quaternion division algebras Q
over a ﬁeld F of characteristic
diﬀerent from 2 is not a division algebra, then there exists a quadratic ex-
tension L of F that embeds as a subﬁeld in Q
and in Q
; see [6, (16.29)].
The same property holds in characteristic 2, with the additional condition
that L/F is separable: this was proved by Draxl , and several proofs have
been proposed: see [3, Theorem 98.19], and  for a list of earlier refer-
Our purpose in this note is to extend the Albert–Draxl theorem by substi-
tuting for the tensor product of two quaternion algebras the corestriction of a
single quaternion algebra over a quadratic extension.
Karim Johannes Becher was supported by the FWO Odysseus Programme (project Explicit
Methods in Quadratic Form Theory), funded by the Fonds Wetenschappelijk Onderzoek
– Vlaanderen. Jean-Pierre Tignol acknowledges support from the Fonds de la Recherche
Scientiﬁque–FNRS under Grants Nos. J.0014.15 and J.0149.17.