# Inertia coprime to degree implies Inertia cyclic?

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.

• Perhaps I should clarify. I'm not looking for a proof so much as good references to use in this proof. – Christine McMeekin Oct 22 '18 at 8:16