 Finite extensions of the complex Laurent series field

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.