Exercise 3, § 1

a) Let $K$ be a maximal ordered field, endowed with the topology $\mathscr{T}_0(K)$, which is compatible with its field structure (Chapter IV, § 3, Exercise 2). Let

be a polynomial in $K[X]$ which has a simple root $\alpha$ in $K$. Show that, for every element $\varepsilon > 0$ in $K$, there exists $\eta > 0$ in $K$ such that every polynomial $g(X) = b_0 X^n + b_1 X^{n-1} + \cdots + b_n$, for which $|a_i - b_i| \leq \eta$ for all $i$, has exactly one simple root $\beta$ such that $|\beta - \alpha| \leq \varepsilon$. (Expand $f$ and $g$ in powers of $X - \alpha$.) Give an upper bound for $\eta$ in terms of the $a_i, \alpha$ and $\varepsilon$.

b) Deduce that, if $K$ is a maximal ordered field, its completion $\hat{K}$ with respect to $\mathscr{T}_0(K)$ (which is a field with a natural order structure: Chapter IV, § 4, Exercise 2) is a maximal ordered field.

c) Let $K_0$ be an ordered field and let $S = K_0((X))$ be the field of formal power series in one indeterminate over $K_0$; linearly order $S$ by taking the elements $\geq 0$ in $S$ to be $0$ and the formal power series whose term of lowest degree has coefficient $> 0$. Show that $S$, endowed with the topology $\mathscr{T}_0(S)$, is complete, but that $S$ is not a maximal ordered field (show that the polynomials $Y^p - X$ of $S[Y]$ are irreducible for each integer $p > 1$).

d) In the example of c), take $K_0 = \mathbf{R}$. Let $K$ be a maximal ordered algebraic extension of $S$; show that $K$ is not complete in the topology $\mathscr{T}_0(K)$. (Embed $S$ in the field $E$ of formal power series with well-ordered exponents, ordered in the same manner as $S$, and embed $K$ in a maximal ordered extension $\Omega$ of $E$. Observe that $E$ is complete, and give an example of a Cauchy sequence in $K$ which converges to an element $f \in E$ which is not algebraic over $S$; choose $f$ so that there is an infinite number of $S$-isomorphisms of $E$ into an algebraic closure of $\Omega$, such that the images of $f$ under these isomorphisms are all distinct.)