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Let $K$ be a number field and let $\mathfrak{p}$ be a prime of $K$ co-prime to 2. Let $L/K$ be a Galois extension of degree a power of 2. Let $I$ denote the inertia group for $\mathfrak{p}$ relative to $L/K$.

Prove (or disprove and salvage if possible) that $I$ is cyclic. (Do I need additional assumptions on $K$ to ensure that $I$ is cyclic?)

This should just be standard algebraic number theory coming from tame ramification at $\mathfrak{p}$, but I am having trouble finding a reference that makes this clear.

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  • $\begingroup$ Perhaps I should clarify. I'm not looking for a proof so much as good references to use in this proof. $\endgroup$ – Christine McMeekin Oct 22 '18 at 8:16
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Corollary 7.59 in Chapter 7 of Milne's Algebraic Number Theory is what I was looking for. Thanks!

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