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Forms over fields and Witt’s lemma. (English) Zbl 1448.11076
Summary: We give an overview of the general framework of forms of A. Bak [On modules with quadratic forms. Algebr. K-Theory Geom. Appl., Lect. Notes Math. 108, 55–66 (1969; Zbl 0192.37202)], J. Tits [Invent. Math. 5, 19–41 (1968; Zbl 0155.05202)] and C. T. C. Wall [Proc. Camb. Philos. Soc. 67, 243–250 (1970; Zbl 0197.31103)], when restricting to vector spaces over fields, and describe its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt’s lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.
MSC:
11E04 Quadratic forms over general fields
15A63 Quadratic and bilinear forms, inner products
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