Quadratic forms and valued fields

Abstract: In this talk, we introduce quadratic forms over fields, and briefly discuss some basic properties. We will then zoom in on the behaviour of quadratic forms over fields carrying a valuation, especially a henselian valuation, like the field of $p$-adic numbers $\mathbb{Q}_p$, or the field of formal Laurent series $\mathbb{R}((T))$. We will see that quadratic forms over a henselian valued field van often be well understood via the valuation’s value group and the quadratic form theory of its residue field, a notion often referred to as Springer’s Theorem. In the exposition we will try wherever feasible to work over fields of arbitrary characteristic, where subtleties may arise which are not visible in fields of other characteristics.

This talk was part of a lecture series at the summer school Algebra and Arithmetic (ALGAR), which in 2023 had as its topic Local-global principles for quadratic forms. The slides of this talk may be found here.