Exercise 2, § 5

Let K be a (commutative) field and $\mathscr{T}$ a non-discrete Hausdorff topology on K compatible with the ring structure on K. For the topology $\mathscr{T}$ to be defined by a valuation on K or an absolute value on K, it is necessary and sufficient that $\mathscr{T}$ be locally retrobounded (General Topology, Chapter 111, § 6, Exercise 22. If there exist in K topologically nilpotent elements, use Exercise 22 (d) of General Topology, Chapter III, § 6 and Exercise 13 of General Topology, Chapter IX, § 3. In the opposite case, use Exercise 22 (f) of General Topology, Chapter III, § 6).