Etale covers of $\mathbb{G}_{m,k}$ in char 0

Question

Let $k$ be a field of characteristic 0. It seems it is well known that étale covers of $\mathbb{G}_{m,k}=\operatorname{Spec}(k[T^{\pm 1}])$ are in bijection with étale covers of $\operatorname{Spec}(k(\!(T)\!))$.

I'd like to understand this by myself, as much as I can. This means I'm here mainly for hints.

What is the good "language" to study étale covers of $\mathbb{G}_m$? I know that, in characteristic 0, connected étale covers of $\mathbb{A}^1_k$ are trivial, i.e. they are the same as connected étale covers of $k$, i.e. they corresponds to finite (separable, but everything is separble in char 0) extensions of $k$.

This is proved, for instance in Galois Theory for Schemes by H.W. Lenstra (6.23) via ramification theory of valuations, with references to some chapters of Intro. to theory of algebraic functions in one variable by Chevalley.

I don't know much of valuation theory and I'm wondering if this "language" may help me to classify étale covers of $\mathbb{G}_m$. I believe it would help me indeed, but maybe there are some other kind of arguments that would solve my problem.

(I apologize for the tag "étale cohomology", maybe it's an overkill, but it was the only tag with the word "étale")

Answer 1

You may try to argue along the following lines.

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If you have read through the notes of Lenstra which you cited, then it should be clear that the question is the following:

Let $k$ be a field of characteristic $0$, and write $K$ for $k(t)$.

There is a bijection between:

• finite extensions $L/K$ which are unramified at all places of $K$, except for the infinite place $v_\infty$ and the place $v_0$ at $t = 0$;
• finite extensions of $k((t))$.

As a first remark, if $L/K$ is unramified outside $v_0$ and $v_\infty$, then there can be only one place of $L$ above $v_0$ (same for $v_\infty$). This follows from the Riemann-Hurwitz formula.

Therefore in one direction we have a well-defined map: taking $E = L \otimes_K k((t))$ gives us the completion of $L$ at the unique place above $v_0$, and this is a finite extension of $k((t))$.

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It remains to see that this map is a bijection.

There is, I believe, middle steps:

(1) Every finite extension of $k((t))$ is of the form $k'((t^{1/n}))$, where $k'/k$ is a finite extension and $n$ is a positive integer.

And on the other side:

(2) Every finite extension $L/K$ which is unramified outside $v_0$ and $v_\infty$ is of the form $k'(t^{1/n})$, where $k'/k$ is a finite extension and $n$ is a positive integer.

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Both (1) and (2) are not so obvious, though. Perhaps it's worth pointing out that they are all specially for function fields: the number field counterparts are far from being true.