A note on quadratic forms

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For a field extension (Formula presented.) we consider maps that are quadratic over (Formula presented.) but whose polarisation is only bilinear over (Formula presented.). Our main result is that all such are automatically quadratic forms over (Formula presented.) in the usual sense if and only if (Formula presented.) 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.

TidsskriftBulletin of the London Mathematical Society
Udgave nummer5
Sider (fra-til)1803-1818
StatusUdgivet - 2024

Bibliografisk note

Funding Information:
We heartily thank Julius Frank for an afternoon of working through the example , Lukas Brantner for a very useful exchange about infinite purely inseparable extensions, Andy Senger for a discussion about the relative Frobenius, and Thomas Nikolaus for hosting FH and MR during a visit to M\u00FCnster, where this note took shape. AK and FH were supported by the German Research Foundation (DFG) via the collaborative research centres \u201CGeometry: Deformations and Rigidity\u201D (grant no. SFB 1442\u2013427320536) at the University of M\u00FCnster and \u201CIntegral structures in Geometry and Representation theory\u201D (grant no. TRR 358\u2013491392403) at the University of Bielefeld, respectively. AK was furthermore supported by the cluster \u201CMathematics M\u00FCnster: Dynamics\u2013Geometry\u2013Structure\u201D under grant no. EXC 2044\u2013390685587. MR was supported by the Danish National Research Foundation (DNRF) through the \u201CCopenhagen Center for Geometry and Topology\u201D under grant no. DNRF151.

Publisher Copyright:
© 2024 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society.

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