= Solution
Normalize $v_K$ to have value group $\mathbb Z$, and extend it to an algebraic closure. Suppose $\alpha$ is a root of a monic <Eisenstein polynomial> of degree $n$. A root cannot have nonpositive valuation: the leading term would then have uniquely smallest valuation. If $w=v_K(\alpha)>0$, the constant term has valuation one, all the other nonleading terms have valuation strictly greater than one, and the leading term has valuation $nw$. Cancellation forces $nw=1$. The value group of $K(\alpha)$ is $(1/e)\mathbb Z$, so $n\mid e$. Since $e\leq[K(\alpha):K]\leq n$, equality holds throughout. Thus the polynomial is irreducible and the extension is <totally ramified>.
Conversely, let $L/K$ be <totally ramified> of degree $n$, and choose a <uniformizer> $\pi_L$. Its extended $v_K$ value is $1/n$. The same denominator argument gives $[K(\pi_L):K]\geq n$, so $K(\pi_L)=L$. Every conjugate has valuation $1/n$, by the <unique extension of an absolute value to a finite extension> of a complete field. The nonleading coefficients of its minimal polynomial are elementary symmetric expressions in these conjugates, so their valuations are positive, hence at least one because the coefficients lie in $K$. The constant coefficient is their product and has valuation exactly one. \b[Its minimal polynomial is an <Eisenstein polynomial>, and $\boxed{L=K(\pi_L)}$.]
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