Abstract
For a field extension L/K we consider maps that are quadratic over L but whose polarisation is only bilinear over K. Our main result is that all such are automatically quadratic forms over L in the usual sense if and only if L/K is formally unramified. In particular, this shows that over finite and number fields, one of the axioms in the standard definition of quadratic forms is superfluous.
| Original language | English |
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| Number of pages | 16 |
| Journal | Bulletin of the London Mathematical Society |
| Early online date | 1 Apr 2024 |
| DOIs | |
| Publication status | E-pub ahead of print - 1 Apr 2024 |
Bibliographical note
Open Access via the Wiley AgreementAK and FH were supported by the Ger-manResearchFoundation(DFG)via thecollaborative researchcentres“Geometry:Deformationsand Rigidity” (grant no. SFB 1442–427320536) at the University of Münster and “Integral struc-tures in Geometry and Representation theory” (grant no. TRR 358–491392403) at the Universityof Bielefeld, respectively. AK was furthermore supported by the cluster “Mathematics Münster:Dynamics–Geometry–Structure” under grant no. EXC 2044–390685587. MR was supported by theDanish National Research Foundation (DNRF) through the “Copenhagen Center for Geometryand Topology” under grant no. DNRF151