= Finite extensions of the complex Laurent series field
{title2=$L\cong\mathbb C((t^{1/n})),\quad[L:\mathbb C((t))]=n$}
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 $1,s,\ldots,s^{n-1}$ 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 $s\mapsto\zeta_ns$, giving a cyclic Galois group.
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