Exercise 20, § 8

Let K be a field of characteristic $\neq 2$ which is Henselian with a valuation $v$; let $\xi \mapsto \bar{\xi}$ be an involutive automorphism of K such that $v(\bar{\xi}) = v(\xi)$ and let $\Phi$ be a non-degenerate Hermitian form on a finite-dimensional vector space E over K.

(a) Let $U = (\alpha_{ij})$ be the matrix of $\Phi$ with respect to a basis $(e_i)$ of E; show that there exists an element $\lambda$ of the order group of $v$ such that, if $\Phi'$ is another Hermitian form on E whose matrix $U' = (\alpha'_{ij})$ with respect to $(e_i)$ satisfies the conditions $v(\alpha_{ij} - \alpha'_{ij}) > \lambda$ for every ordered pair $(i, j)$, then $\Phi$ and $\Phi'$ are equivalent. (Argue as in Algebra, Chapter IX, § 6, Exercise 6, using Exercise 14 (a) above.)

(b) Suppose that $\xi \mapsto \bar{\xi}$ is the identity (hence $\Phi$ is a symmetric form) and that the valuation $v$ is discrete, normed and such that $v(2) = 0$; let x be a uniformizer for $v$. Show that $\Phi$ is characterized, up to equivalence, by its index $v$ and by two symmetric bilinear forms $\Psi_1, \Psi_2$ of index 0 on vector spaces $k^r$ and $k^s$ respectively, where k is the residue field of $v$ and $r + s = n - 2$. (With the aid of a Witt decomposition, reduce it to the case where $\Phi$ is of index 0; with the aid of Exercise 19, show that, for all $i \geq 0$, the set $M_i$ of $x \in E$ such that $v(\Phi(x, x)) \geq i$ is a module over the ring A of the valuation $v$. Show that, if $x \in M_i, y \in M_{i+1}$, then $v(\Phi(x, y)) \geq i + 1$ and, by taking quotients, symmetric bilinear forms can therefore be derived from $\Phi$ on the vector k-spaces $M_0/M_1$ and $M_1/M_2$. Use Exercise 6 (c) of § 3 and the fact that the equation $\xi^2 = \alpha$ has a solution in A for $a \equiv 1 \pmod{\pi}$.) Consider the case where k is finite (cf. Algebra, Chapter IX, § 6, Exercise 4).