Lemma 12.1. Assuming that:

Then the Trace form

\begin{array}{l} (∙, ∙) : R \times R & \to R \\ (x, y) & \mapsto {Tr}_{R ∕ k} (x y) (: = {Tr}_{k} (mult (x y))) \end{array}

is non-degenerate if and only if $R = k_{1} \times \dots \times k_{r}$ where $k_{i} ∕ k$ is a finite separable extension of $k$ .