An explicit formula for Larmour's decomposition of hermitian forms

Abstract

Let $K$ be a complete discretely valued field whose residue field has characteristic different from $2$. Let $(D,\sigma)$ be a $K-$division algebra with involution of the first kind, and $h$ be a $K-$anisotropic $\epsilon$-hermitian form over $(D,\sigma)$. By a theorem due to Larmour, there is a decomposition $h=h_0 \perp h_1$ such that the elements in a diagonalization of $h_0$ are units, the elements in a diagonalization of $h_1$ are uniformizers, and $h_0$, $h_1$ are determined uniquely up to $K-$isometry. In this paper, we give an explicit description of the elements in the diagonalization of $h_0$ and $h_1$ in the case of quaternion algebras. Then we will derive explicit formulas for Larmour's isomorphism of Witt groups.