Exercise 2, § 4

Let $K$ be a non-discrete Hausdorff topological division ring (or field).

a) Let $E$ be a left vector space of dimension $n$ over $K$. If $(a_i)_{1 \leq i \leq n}$ is a basis of $E$, and if we transport to $E$ the topology of $K^n$ (the product of the topologies of the factors) by means of the bijective linear mapping

then the topology so defined on $E$ is independent of the basis $(a_i)$ chosen, is compatible with the additive group structure of $E$, and is such that the mapping $(t, x) \to tx$ of $K \times E$ into $E$ is continuous. If $F$ is a vector subspace of $E$, the topology induced on $F$ by that of $E$ is the same as the topology defined on $F$ by starting from an arbitrary basis of $F$, as above; $F$ is closed in $E$, and $\mathbf{C}F$ is dense in $E$ unless $F = E$.

b) Generalize Propositions 4, 5 and 6 of Chapter VI, § 1 to the division ring $K$. (To show that the mapping $X \to X^{-1}$ is continuous in the neighbourhood of every nonsingular square matrix of order $n$ over $K$, when $K$ is not commutative, use induction on $n$: considering a neighbourhood of the unit matrix $I_n$ of order $n$, note that every matrix $X$ sufficiently near to $I_n$ can be put in the form

where $I_{n-1}$ is the unit matrix of order $n - 1$ and $\gamma$ is a nonsingular matrix of order $n - 1$.)

c) If $K$ is commutative and $E$ is an algebra of rank $n$ over $K$, then the topology of $E$ is compatible with its ring structure. Furthermore, if $E$ has an identity element, the group $G$ of units of $E$ is open and dense in $E$, and the topology induced on $G$ by that of $E$ is compatible with the group structure of $G$. (If $e$ is the identity element of $E$, and if $x$ is a unit of $E$, then each of the equations $xy = e, yx = e$ has one and one only solution $y$; for each of these equations consider the equivalent system of $n$ linear equations which give the components of $y$ with respect to a basis of $E$.)

d) If $K$ is a non-complete field and $E$ an algebra of rank $n$ over $K$, and if the completion $\hat{K}$ of $K$ is a field, then the completion $\hat{E}$ of the algebra $E$ is that obtained by extending the ring of operators of $E$ to $\hat{K}$. Hence construct examples of a topological division ring whose completion is not a division ring.