Exercise 2, § 3

a) On the polynomial ring $A = \mathbf{R}[X]$, consider the linearly ordered ring structure for which the elements $\geqslant 0$ are $0$ and all polynomials $\neq 0$ whose leading coefficient is $> 0$. Show that the topology $\mathfrak{T}_0(A)$ is not compatible with the ring structure of $A$.

b) If $K$ is an ordered field the topology $\mathfrak{T}_0(K)$ is compatible with the ring structure of $K$, and the bounded sets for this ring topology (Chapter III, § 6, Exercise 12) are the bounded sets for the order structure. The topology $\mathfrak{T}_0(K)$ is locally retrobounded (Chapter III, § 6, Exercise 22) and in particular is compatible with the field structure of $K$; moreover the completion $\hat{K}$ of $K$ with respect to this topology, which is a field (\emph{loc. cit.}) is canonically endowed with the structure of an ordered field [§ 1, Exercise 3 b)].

c) On the field $K = \mathbf{Q}(\sqrt{2})$, considered as an ordered subfield of $\mathbf{R}$, consider the topology $\mathfrak{T}$ which is such that $(x, y) \to x + y \sqrt{2}$ is a homeomorphism of $\mathbf{Q}^2$ onto $K$, $\mathbf{Q}^2$ carrying the product topology. This topology $\mathfrak{T}$ is finer than $\mathfrak{T}_0(K)$ and is compatible with the field structure of $K$; but the completion of this topological field is not a field.

d) The field $K = \mathbf{Q}(\sqrt[3]{2})$, considered as a vector space over $\mathbf{Q}$, is the direct sum of the vector subspace $G'$ generated by $1$ and $\sqrt[3]{2}$, and the vector subspace $G''$ generated by $\sqrt[3]{4}$. Consider the topologies on $G'$ and $G''$ induced by the topology of $\mathbf{R}$; let $\mathfrak{T}$ be the topology on $K$ which is the product of the topologies of $G'$ and $G''$. Show that $\mathfrak{T}$ is compatible with the additive group structure of $K$ and is finer than $\mathfrak{T}_0(K)$ ($K$ being considered as an ordered subfield of $\mathbf{R}$) but is not compatible with the ring structure of $K$.