Exercise 1, § 9

(a) Let K be a field (not necessarily commutative) and $\mathscr{T}$ a locally compact topology on K compatible with the ring structure on K; show that $\mathscr{T}$ is compatible with the field structure on K. (Use the Theorem of R. Ellis (General Topology, Chapter X, § 3, Exercise 25) or argue directly reproducing the proofs of Integration, Chapter VII, § 1, no. 10 and those of Proposition 1 of this paragraph.)

(b) Give an example of a locally compact topology on a commutative field K, which is compatible with the additive group structure on K, but not with its ring structure. (Take K to be the field of fractions of a compact integral domain A (A being for example a ring of formal power series $k[[X, Y]]$, where $k$ is finite) and take as fundamental system of neighbourhoods of 0 in K the neighbourhoods of 0 in A.)