  Let $p$ be a prime and let $K$ be a finite extension of the field ${\bf Q}_p$
of $p$-adic numbers such that the group ${}_pK^\times$ has order $p$. The ${\bf
F}_p$-space $K^\times\!/K^{\times p}$ carries a natural filtration coming from
the valuation on $K$, and a natural bilinear pairing coming from the
reciprocity isomorphism for the exponent~$p$. We determine the orthogonal
filtration for this pairing. We also prove the analogous result for $p$-fields
of characteristic $p$.
