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Fix a prime $p$ and consider the equation $X^p-X-t^{-1}$ over $\mathbb F_p((t))$, the field of formal Laurent series over $\mathbb F_p$. What is the Galois group of this equation?

After fumbling around with power series for a while, I found that $\sum_{n=1}^\infty t^{1/p^n}$ is a root of this equation. I have some guesses for what the Galois group is but not seeing anything concrete.

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These are called Artin-Schreier extensions. Everything you want to know is at Wikipedia or by searching on that term.

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