  In this article, we prove several transfer principles for the cohomological
dimension of fields. Given a fixed field $K$ with finite cohomological
dimension $\delta$, the two main ones allow to:
  - construct totally ramified extensions of $K$ with cohomological dimension
$\leq \delta - 1$ when $K$ is a complete discrete valuation field;
  - construct algebraic extensions of $K$ with cohomological dimension $\leq
\delta-1$ and satisfying a norm condition.
  We then apply these results to Serre's conjecture II and to some variants for
fields of any cohomological dimension that are inspired by conjectures of Kato
and Kuzumaki. In particular, we prove that Serre's conjecture II for
characteristic $0$ fields implies Serre's conjecture II for positive
characteristic fields.
