Upper bound on exact power of wild prime that divides the different

Question

Let $K$ be a number field and let $p\in \mathbb{Z}$ be a prime. Suppose that $p\mathcal{O}_K=Q^eI$, with $\gcd(Q, I)=1$ and let $\mathcal{D}_{K/\mathbb{Q}}$ be the different ideal of $K$. We know that $Q^{e-1}\mid \mathcal{D}_{K/\mathbb{Q}}$ and if $Q$ is tamely ramified over $p$, i.e. $p\not\mid e$, then $Q^e\not\mid \mathcal{D}_{K/\mathbb{Q}}$. Suppose now that $Q$ is wild over $p$, i.e. $p\mid e$, and let $k$ be the exact power of $Q$ that divides $\mathcal{D}_{K/\mathbb{Q}}$. If we know $p$ and $e$, can we give an upper bound on $k$?

Answer 1

Yes, there is an upper bound depending only on $p$ and $e=p^n$, and it’s achieved in the extension $K=\Bbb Q(p^{1/e})$.

This is a purely local question, and I’m going to pretend that you asked it for wildly ramified extensions of $\Bbb Q_p$, though the translation to $\Bbb Q$ is almost automatic. First let’s look at the above extension:

We have $\pi$, a root of $X^{p^n}-p$, whose derivative is $p^nX^{p^n-1}$, substitute $\pi$ and get $p^n\pi^{p^n-1}=\delta$, a number with $v_\pi(\delta)=np^n+p^n-1=(n+1)e-1$. I say that this is as big as your number $k$ can get.

Our general extension $K\supset\Bbb Q_p$ is to be totally wildly ramified, so of degree $p^n=e$, with a prime element $\pi$ that’s root of an Eisenstein polynomial $F(X)\in\Bbb Q_p[X]$. Say $F=X^{p^n}+\sum_0^{e-1} c_jX^j$, with $\delta=F'(\pi)=p^n\pi^{p^n-1}+\sum_0^{p^n-2}(j+1)c_{j+1}\pi^j$. When you ask for the $v_\pi$-value of this, you notice that no two monomials in the expression have the same value, since all are incongruent modulo $p^n=e$: the numbers $v_\pi(c_{j+1})$ are in $\Bbb Q_p$ and thus are divisible by $e=p^n$. So the minimum of the $v_\pi$-values of these monomials may well be less than that of $p^n\pi^{p^n-1}$, but since this monomial might also dominate, as it did in my example, that is the biggest possible value for $k$.