Let $K$ and $K'$ be local field, and $v$ and $v'$ be it's valuation. Suppose $K'/K$ is unramified, then, for all $u'∈K'$,there exists $u∈K$ such that $u/u'∈R'^×$.
My attempt: To probe this, $v'(u/u')＝v'(u)-v'(u')＝v(u)-v'(u')$, so, let $u'$ be $v'(u)$-th power of prime element of $K$, then, $v'(u/u')＝0$ and this $u$ is what we wanted.
Is my attempt correct?
This question is from Silverman's 'the arithmetic of elliptic curves', p198.
Thank you in advance.