Exercise 6, § 3

Let $A$ be a Noetherian local integral domain, $m$ its maximal ideal and $K$ its field of fractions; suppose that $A$ is complete with the $m$-adic topology. Consider on $A$ a linear topology $\mathscr{T}$ which is finer than the $m$-adic topology. Show that, if $\mathscr{T}'$ is the linear topology on $K$ for which a fundamental system of neighbourhoods of 0 consists of the neighbourhoods of 0 in $A$ with the topology $\mathscr{T}$, $\mathscr{T}'$ is compatible with the field structure on $K$ and $K$ is complete with the topology $\mathscr{T}'$.