Arch. Math. 111 (2018), 135–143
2018 Springer International Publishing AG,
part of Springer Nature
published online May 30, 2018
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.