Exercise 9, § 8

Let K be a field, L a finite algebraic extension of K, $v'$ a valuation on L, A' its ring, $v$ the restriction of $v'$ to K and $A = A' \mathbf{A} K$ its ring.

(a) Suppose that A is Henselian (Exercise 6); show that the product $e(v'/v)f(v'/v)$ divides $n = [L:K]$ and that the quotient $n/e(v'/v)f(v'/v)$ is a power of the characteristic exponent of the residue field k of v. (Reduce it to the case where L is a Galois extension of K; use Exercises 6 (e) and 7 and Chapter V, § 2, Theorem 2 and Proposition 5.)

(b) Suppose further that $n$ is not divisible by the characteristic exponent of the residue field k of v and that $f(v'/v) = 1$. Show that, for every integer m equal to the order of an element y of the group $\Gamma_{v'}/\Gamma_v$, there exists $x \in L$ such that the class of $v'(x)$ mod. $\Gamma_v$ is equal to y and $x^m \in K$.