8 KARIM JOHANNES BECHER
Theorem 4.2 the ramification sequence
ρ=∂({t
2
+ (a+ 1)t+a, a}+{t
2
+at+a, ab})
satisfies the claim in (a).
To show (b), we consider theE(t)-biquaternion algebra
B= (t
2
+ (a+ 1)t+a, a)⊗E(t)(t
2
+at+a, ab).
For anyf, g∈E(t)
×
such thatB⊗E(t)(f, g) can be defined overE, we would
obtain thatρ=∂({f, g}), in contradiction to (a). Therefore there exists noE(t)-
quaternion algebraQsuch thatB⊗E(t)Qcan be defined overE. In particular,B
does not contain anyE-quaternion algebraQ
′
, because otherwise the centraliser
ofQ
′
E(t)
inBwould be anE(t)-quaternion algebraQsuch thatB⊗E(t)Qis
defined overE. SinceBdoes in particular not containM2(E), it follows thatB
is a division algebra.
Acknowledgments.This work was supported by the FWO Odysseus Programme
(projectExplicit Methods in Quadratic Form Theory), funded by the Fonds
Wetenschappelijk Onderzoek – Vlaanderen.
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Universiteit Antwerpen, Departement Wiskunde–Informati ca, Middelheimlaan 1,
2020 Antwerpen, Belgium.
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