Exercise 4, § 5

Let $v$ be a valuation on a field K, A the ring of $v$ and m its maximal ideal.

(a) For the topology defined by $v$ on A to be identical with the m-adic topology, it is necessary and sufficient that A be a field or a discrete valuation ring.

(b) For the topology defined by $v$ to make A into a strictly linearly compact ring (Chapter 111, § 2, Exercise 21), it is necessary and sufficient that A be a field or a discrete valuation ring (use (a) and Chapter III, § 2, Exercise 22 (a)).