Exercise 14, § 8

(a) For a field $K$ to be Henselian with a valuation $v$, it is necessary and sufficient that $K$ be the greatest separable algebraic extension of $K$ contained in $\hat{K}$ and that $\hat{K}$ be Henselian. (To see that the condition is necessary, use Exercise 12 (b); to show that it is sufficient, observe that, to verify condition (H) of Chapter III, § 4, Exercise 3, we may confine our attention to the case where $P$ is separable over $K$, noting that, if $Q$ is an irreducible factor of $P$ in $\hat{K}[X]$ such that $Q^{p^e}$ belongs to $K[X]$, then $Q$ is a g.c.d. of $P$ and $Q^{p^e}$ in $\hat{K}[X]$.) Consider the case where $v$ is of height 1 (cf. Exercise 6 (b)).

(b) Deduce from (a) that there exists in the $p$-adic field $\mathbf{Q}$, a countable subfield which is Henselian with the $p$-adic valuation.

(c) Give an example of a field $K$ which is Henselian with a discrete valuation which is not complete and such that $\hat{K}$ is a finite radicial extension of $K$ (cf. Chapter V, § 1, Exercise 20).