Exercise 40, § 2

Let K be an ordered field. Make the field of formal power series $E = K((X))$ an ordered field by taking the elements $> 0$ to be the formal power series $\sum_{n=h}^{\infty} r_n X^n$ whose nonzero coefficient of least degree is $> 0$. a) Show that if $L = K(X)$ is the subfield of E consisting of rational functions in one indeterminate, then $E = \tilde{L}$ (Ex. 36) up to isomorphism of ordered fields (observe that for every element $a > 0$ of L there exists an integer $n > 0$ such that $0 < X^n < u$). b) Suppose that K is maximal ordered. Show that the union F of the fields $K((X^{1/n}))$ (ordered in the same way as E) is an ordered algebraic extension of E which is maximal ordered (use Ex. 2 of V, p. 150). c) Show that the field $\tilde{F}$ (Ex. 36) is the field of formal series $\sum \alpha_r X^r$, where r runs through the set $\mathbf{Q}$ of rational numbers, and where $a_r \in K$ and the set of r such that $a_r \neq 0$ is an increasing sequence which is either finite or tends to $+\infty$; order this field in the same manner as E. In particular $\tilde{F} \neq F$.