Let $R$ be a complete DVR with fraction field $K$. Let $d\geq 1$ be an integer. Does $K$ have only finitely many extensions of degree $d$?
If $K=\mathbb Q_p$ the answer is yes. Is this a general fact about complete DVRs?
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1If you take a field $k$ which has infinitely many extensions of degree $d$ and take $K$ to be the field of Laurent series with coefficients in $k$, what happens? – Mariano Suárez-Álvarez Dec 14 '16 at 16:27
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1@MarianoSuárez-Álvarez Ow right, I see. Then there must be some condition on the residue field. What if we assume the residue field (of the valuation ring) to have only finitely many extensions of bounded degree? – Harry Dec 14 '16 at 16:33
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1See this thread for some discussion that is AFAICT relevant to your topic. Note that $\Bbb{Q}_p$ and $\Bbb{F}_p((X))$ behave differently in this respect. – Jyrki Lahtonen Dec 15 '16 at 15:31