Given a local field $K$ how does one obtain the maximal unramified extension of $K$?

Question

Let $K$ be a local field complete with respect to a discrete valuation $v$. I am interested in learning what do we need to add to $K$ to make it the maximal unramified extension of $K$. Any comments are appreciated. Thank you.

Answer 1

Suppose that the residue field of $K$ has cardinality $q=p^f$ for some prime $p$. Then the only finite unramified extension $K_n$ of $K$ of degree $n$ (once we've fixed an algebraic closure etc) is obtained by adjoining to $K$ the roots of $x^{q^n}-x$. Now it turns out this extension is an abelian extension with Galois group $\mathbf{Z}/n\mathbf{Z}$. The maximal unramified extension is obtained by taking the union, $K=\bigcup_{n \geqslant 1} K_n$, and its Galois group is $\hat{\mathbf{Z}}$, the profinite completion of $\mathbf{Z}$.