  For a field $E$ of characteristic different from $2$ and cohomological
$2$-dimension one, quadratic forms over the rational function field $E(X)$ are
studied. A characterisation in terms of polynomials in $E[X]$ is obtained for
having that quadratic forms over $E(X)$ satisfy a local-global principle with
respect to discrete valuations that are trivial on $E$. In this way new
elementary proofs for the local-global principle are achieved in the cases
where $E$ is finite or pseudo-algebraically closed. The study is complemented
by various examples.
