Exercise 12, § 1

Let K be a discrete field, E the division ring of fractions of the ring of formal series $A = K[[X, Y]]$ in two indeterminate variables on K (A, IV, p. 36). For every $n \geq 0$, let $V_n \subset A$ be the set of formal series of total degree at least equal to n. Show that in E, the sets $V_n$ form a fundamental system of neighbourhoods of 0, for a topology compatible with the vector space structure of E (over K), for which E is metrisable and complete; if further K is a finite field, then E is locally compact. Show that the K-bilinear mapping $(u, v) \mapsto uv$ of $E \times E$ in E is continuous at the point (0, 0) but that there exists $u_0 \in E$ such that $v \mapsto u_0 v$ is not continuous in E (for example $u_0 = 1/X$).