Remark

Note that it is sufficient in this proof to suppose that $A$ is complete.

We shall see in § 9 that a field $K$ fulfilling the conditions of Proposition 2 admits a centre which is either a finite algebraic extension of a $p$-adic field, or a field $\mathbf{F}_q((T))$ of formal power series over a finite field; moreover $K$ is of finite rank over its centre.