Suppose $K/F$ is a finite extension with Galois group $Gal(K/F)$. Prove that there is an maximal intermediate field $L$ of odd degree, in the sense that $[L:F]$ is odd, and any other odd degree intermediate field must be contained as an isomorphic copy inside $L$.
If $|Gal(K/F)|$ contains an odd prime factor, then by the Sylow theorem, we can find a subgroup of odd order. By the fundamental theorem of Galois theory, this means that there exists an intermediate field of odd degree. However, I do not know how to find one that satisfies the second condition. I think that finding an odd order subgroup that is contained in every other odd order subgroup would work since fundamental theorem of Galois theory is inclusion reversing, but I do not now if this is possible or the correct way to think about this. Thanks for the help!
The fixed field of a 2-Sylow subgroup. Any odd degree intermediate field is fixed by a subgroup of odd index, hence by a subgroup having a 2-Sylow of the same size as and conjugate to the one we picked.