Exercise 19, § 6

a) Let $K$ be a division ring endowed with a Hausdorff topology $\mathfrak{T}$ compatible with the ring structure of $K$. Show that there is a neighbourhood $V$ of $o$ in $K$ such that $(CV) \cup (CV)^{-1} = K^*$ [this is equivalent to $V \cap (V \cap K^*)^{-1} = \varnothing$].

b) Suppose moreover that $\mathfrak{T}$ is not discrete. Show that if $U$ is any neighbourhood of $o$ in $K$ and if $U^*$ denotes $U \cap K^*$, then

(*) It can be shown that a compact ring with an identity element is necessarily totally disconnected.