Suppose $p$ is prime and $p$ divides $e$ divides $n$. Typically up to isomorphism there are a lot of wildly ramified extensions of $\mathbb{Q}_p$ which have degree $n$ and ramification index $e$. Krasner's Mass Formula gives us an exact number of extensions, but I was wondering whether anyone knew if it was possible to get a definite lower bound on the number of degree $n$ extensions "up to isomorphism" of $\mathbb{Q}_p$ with ramification index $e$.
I am investigating a property which I think should only apply to tamely ramified extensions, but wanted to know whether I could be certain that it does not apply to any wildly ramified extensions. If I could show the number of extensions of $\mathbb{Q}_p$ up to isomorphism with degree $n$ and ramification index $e$ is $> e$, then that would be awesome.
Unfortunately I don't know a lot about bounding a count of extensions up to isomorphism, all I know how to do is to cross-check generating polynomials using Panayi's algorithm to see which extensions are isomorphic.
Perhaps I am out of my depth, but why would you want a mere lower bound when Krasner's formulas already provide an exact count? If I understand correctly, by Krasner's, the expression
$$\large {\frak N}=e+e\sum_{s=1}^{v_p(e)} p^{\Large s+\frac{n}{p^{s-1}}\frac{p^{s-1}-1}{p-1}}\left(p^{\Large \frac{n}{p^s}}-1\right)$$
is the number of isomorphism classes of extensions of ${\bf Q}_p$ of degree $n$ and ramificationin index $e$; I base this off the first introductory paragraph of the paper Enumeration of isomorphism classes of extensions of $p$-adic fields. If $p\nmid e$ (i.e. we're looking at tame extensions) the sum is empty and the count is precisely $e$, but otherwise (if we're considering wild extensions) $p\mid e$ implies the sum is non-empty, and it is clear the terms are all positive so the count is indeed $>e$ as you desire.