I was just wondering given a local field complete with non-archimedean valuation, is the maximal unramified extension always finite or could it be infinite? Any comments are appreciated!
2026-02-22 19:52:06.1771789926
Given a local field is the maximal unramifield extension always finite?
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For a finite extension of $\Bbb Q_p$ the maximal unramified extension is infinite. It is generated by adjoining the $n$-th roots of unity for all $n$ coprime to $p$. The Galois group is the cyclic profinite group $\hat{\Bbb Z}$, and is naturally isomorphic to the Galois group of the algebraic closure of $\Bbb F_p$. This is in all textbooks on local fields, for instance Serre's.