Laurent series field ($k((t))$)

The Laurent series field $k((t))$ consists of formal series ${\sum }_{n\ge N}{a}_n{t}^n$ with finitely many negative-degree terms. Its $t$-adic valuation is the least exponent with nonzero coefficient.

For arbitrary positive integers $e,f$, the displayed extension has degree $ef$, ramification index $e$, and residue degree $f$. If $|t|=p^{-1}$, its unique extended absolute value on a field is $|\sum a_n{u}^n|=p^{-\min \{n:a_n\ne 0\}/e}$. A basis consists of ${\gamma }_i{u}^j$, where the ${\gamma }_i$ form an basis of ${\mathbb{F}}_{p^f}$ and $0\le j<e$. Regrouping a Laurent series by its exponent modulo $e$ proves the basis assertion. When $p$ divides $e$ the extension can be inseparable; the valuation calculation still applies.

A finite extension of the complete discretely valued field $\mathbb{C}((t))$ has residue field $\mathbb{C}$, hence is totally ramified. If $\pi$ is its uniformizer, write $t=u{\pi }^n$. A root of the residue of $u$ lifts by Hensel lemma to an $n$th root $v$ of $u$, since the residue characteristic is zero. Then $s=v\pi$ satisfies $s^n=t$. Distinct valuations modulo $n$ make $1,s,\dots ,s^{n-1}$ linearly independent over the base, so $L=\mathbb{C}((t))(s)$. Expanding in $s$ with constant coefficients identifies it with $\mathbb{C}((s))$. Its automorphisms are $s\mapsto {\zeta }_{n}s$, giving a cyclic Galois group.

Inside an algebraic closure, choose compatible roots $t^{1/n}$. The fields $\mathbb{C}((t^{1/n}))$ exhaust all finite extensions by finite extensions of the complex Laurent series field. Restriction from a degree-$m$ field to a degree-$n$ field for $n\mid m$ sends an exponent modulo $m$ to its reduction modulo $n$. The absolute Galois group is their inverse limit, namely the profinite completion of $\mathbb{Z}$. This is a procyclic group, rather than the direct-limit torsion group of all complex roots of unity.

For an irreducible polynomial $h\in {\mathbb{F}}_p[t]$ of degree $d$, the $h$-adic completion of ${\mathbb{F}}_p(t)$ is isomorphic to ${\mathbb{F}}_{p^d}((T))$. Its residue field is ${\mathbb{F}}_p[t]/(h)$ and $h$ is a uniformizer.