Exercise 1, § 5

Let A be an integral domain, K its field of fractions and $\mathscr{T}$ a linear topology on A (Chapter 111, § 2, Exercise 21).

(a) For the neighbourhoods of 0 under $\mathscr{T}$ to constitute a fundamental system of neighbourhoods of 0 for some topology $\mathscr{T}_K$ compatible with the ring structure on K, it is necessary and sufficient that $\mathscr{T}$ be the topology $\mathscr{T}_u(A)$ (Chapter III, § 2, Exercise 24); then A is a bounded subset with respect to $\mathscr{T}_K$ and $\mathscr{T}_K$ is a locally bounded Hausdorff topology (General Topology, Chapter 111, § 6, Exercises 12 and 20 (e)). For K to be complete (resp. linearly compact, resp. strictly linearly compact (Chapter 111, § 2, Exercise 21)) with $\mathscr{T}_K$, it is necessary and sufficient that A be so with $\mathscr{T}$.

(b) For the topology $\mathscr{T}_K$ (where $\mathscr{T} = \mathscr{T}_u(A)$) to be compatible with the field structure on K, it is necessary and sufficient that the Jacobson radical $\mathfrak{R}(A)$ of A be $\neq 0$.