The maximal unramified extension of $\mathbb{Q}_p$ can be described quite explicitly: add all roots of unity of order prime to $p$. This is done by the correspondence between finite unramified extensions of $\mathbb Q_p$ and finite extensions of $\mathbb F_p$. Similarly, can one find an explicit description for the maximal unramified extension of $\mathbb F_p((t))$?
Maximal Unramified Extension of $\mathbb{F}_p((t))$
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number-theory
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1 Answers
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For $\mathbb{F}_p((t))$, this is in a sense even simpler: the maximal unramified extension is the direct limit over $\mathbb{F}_{p^n}((t))$, $n\in \mathbb{N}$. This is very easy to see, since unramified extensions of a local field are in bijection with extensions of the residue field. In other words, the maximal unramified extension is again generated by roots of unity of order prime to $p$.
It's good fun to try and classify, using class field theory, the maximal tamely ramified extension of $\mathbb{F}_p((t))$.
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0@Alex B. Right. That's it. (There are infinite algebraic extensions of $\mathbb{F}_p((t))$ such that the fundamental equality does not hold. That was confusing me.) – 2012-12-07