Exercise 22, § 6

In a division ring $K$ endowed with a Hausdorff topology $\mathfrak{T}$ compatible with the ring structure of $K$, a set $R$ containing $o$ is said to be retrobounded if $(\mathbf{C}R)^{-1}$ is bounded. $\mathfrak{T}$ is said to be locally retrobounded if there is a fundamental system of retrobounded neighbourhoods of $o$ in $K$ with respect to $\mathfrak{T}$.

a) A locally retrobounded topology $T$ is locally bounded, and is a minimal element in the set of all Hausdorff topologies compatible with the ring structure of $K$; also $\mathfrak{T}$ is compatible with the division ring structure of $K$ [use Exercise 19 a)].

b) Suppose from now on, in this exercise, that $\mathfrak{T}$ is a locally retrobounded topology on $K$. Show that, for every neighbourhood $V$ of $o$ in $K$, $x \to x^{-1}$ is uniformly continuous on $\mathbf{C}V$. Hence show that the completion $\hat{K}$ of $K$ is a topological division ring whose topology is locally retrobounded.

c) Every neighbourhood of $o$ in $K$ (with respect to $\mathfrak{T}$) contains a neighbourhood $V$ of $o$ such that $E(V) = (\mathbf{C}V) \cap (\mathbf{C}V)^{-1}$ is not empty. Show that the three uniformities on $E(V)$, induced by the additive uniformity and the two multiplicative uniformities on $K$, coincide. Deduce that, if $K$ is complete, then $K^*$ is complete with respect to either multiplicative uniformity.

d) Show that if $x$ is an element of $K^*$ then exactly one of the following three statements is true: (i) $x$ is topologically nilpotent; (ii) $x^{-1}$ is topologically nilpotent; (iii) the sequence $(x^n)_{n \in \mathbf{Z}}$ is bounded. [By considering a bounded neighbourhood $U$ of $o$ such that $UU \subset U$

(Exercise 20 d)), show that if o is a cluster point of the sequence $(x^n)_{n \geq 0}$, then $x$ is topologically nilpotent.] If $K$ contains non-zero topologically nilpotent elements, show that the set $B$, consisting of o and all $x \in K^*$ such that $x^{-1}$ is not topologically nilpotent, is a bounded neighbourhood of o in $K$ which is invariant under all inner automorphisms [use Exercise 21 b)]; $B$ then contains all bounded neighbourhoods $U$ of o such that $UU \subset U$; the set of such neighbourhoods which are invariant under all inner automorphisms has a greatest element, which coincides with $B$ if the commutator subgroup of $K^*$ is bounded, in particular if $K$ is a field.

e) If $U$ is a bounded symmetric neighbourhood of o in $K$ such that $UU \subset U$, then there exists $b \in U^*$ such that $K^* = U^* \cup ((U^*)^{-1} b)$. Let $a \in U^*$ be such that $Ua \subset bU$, and suppose that the sequence $(a^{-n})_{n \geq 0}$ is bounded. Show that the set $V$ of all $x \in K$ such that $xU$ is contained in the union of the sets $Ua^{-n}$ is a bounded neighbourhood of o such that $VV \subset V$ and $K^* = V^* \cup (V^*)^{-1}$.

f) Suppose that $K$ contains no non-zero topologically nilpotent elements. Show that there is a bounded neighbourhood of o in $K$ which is a subring $A$ of $K$ such that $K^* = A^* \cup (A^*)^{-1}$. [Begin with a bounded neighbourhood $V$ of o such that $K^* = V^* \cup (V^*)^{-1}$ (see e)]; if $c \in V^*$ is such that $c(V + V) \subset V$, observe that the ring $A$ generated by $V$ is contained in the union of the sets $Vc^{-n}\ (\ n \geq 0$.