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

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 $K$.