The norm principle for type $D_n$ groups over complete discretely valued fields

TL;DR

This paper proves that the norm principle for extended Clifford groups of quadratic forms over a complete discretely valued field follows from its validity over the residue field, under certain assumptions.

Contribution

It establishes the norm principle for extended Clifford groups over complete discretely valued fields assuming it holds over their residue fields.

Findings

Norm principle holds for extended Clifford groups over the field $K$.

Validity over residue field implies validity over the complete discretely valued field.

Results depend on the norm principle holding for all finite extensions of the residue field.

Abstract

Let $K$ be a complete discretely valued field with residue field $k$ with $\mathrm{char}(k)\neq 2$. Assuming that the norm principle holds for extended Clifford groups $\Omega(q)$ for every even dimensional non-degenerate quadratic form $q$ defined over any finite extension of $k$, we show that it holds for extended Clifford groups $\Omega(Q)$ for every even dimensional non-degenerate quadratic form $Q$ defined over $K$.