Exercise 17, § 8

Let K be a Henselian field with a valuation v which is not algebraically closed, and L a subfield of K such that [K: L] is finite. Show that under these conditions L is Henselian with the restriction of v. (In the contrary case, there would exist two extensions $v_1, v_2$ of $v|L$ to an algebraic closure $\Omega$ of K such that the restriction of $v_1$ and $v_2$ to a finite algebraic extension L' of L would not be equivalent. Deduce that there would exist on a suitable finite quasi-Galois extension K' of L containing K, two valuations $v'_1, v'_2$ which are not equivalent and for which K' would be Henselian; using Exercise 15 (c), show that, if $E = L^{p^{-\infty}}$ ($p$ being the characteristic exponent of R), $E \neq \Omega$ but $\Omega$ would be a finite extension of E; complete the argument with the aid of Algebra, Chapter VI, § 2, Exercise 31.)