Let $K/\mathbb{Q}$ be a quadratic number field. Then the discriminant of $K$ is either $d$ or $4d$ for some squarefree integer $d$, so the discriminant is never divisible by an odd square. I am wondering if this can be generalised to number fields of higher degree. In particular, is it the case that there is some $k$ such that the discriminant of a quartic number field is never divisible by an odd $k$th power?
I have checked every quartic number field in the PARI database, and none of them has discriminant divisible by an odd ninth power, so maybe the conjecture holds with $k=9$. This is not particularly strong evidence, though, since the database only contains quartic fields with discriminant below $1,000,000$, and such discriminants can never be divisible by any odd ninth power apart from $3^9$.
This seems like a question whose answer might be well-known. Does anybody know?