Given a complete discrete valuation field $L$, is there always a complete local field $K\subset L$? That is, I assume the residual field of $L$ is non-finite. Can I claim that it contains a complete field with finite residual field with respect to the same valuation?
2026-03-26 13:51:29.1774533089
Given a complete discrete valuation field $L$, is there always a complete local field $K\subset L$?
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Following the lead of the comment by @Torsten Schoeneberg. I point out that $\Bbb R((t))$, the formal Laurent series over the real field, is certainly a complete discrete valuation field, but it contains no field that’s finite over a $\Bbb Q_p$. Why? Because fields of the latter type always have elements whose square is a negative integer, while $\Bbb R((t))$ contains no such element.