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I just read the proof of this theorem that $\mathbb{Q}_p$ has finitely many totally ramified extensions of any degree $n$. The proof uses Krasner's lemma and the compactness of a space which corresponds to Eisenstein polynomials of degree $n$. One then picks a finite subcover which represents every possible such extension.

This proof technique is not very useful if one actually wants to count the number of totally ramified extensions of a particular degree $n$. Does anyone know of any actual methods for computing this?

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Krasner apparently also derived a formula for the number of totally ramified extensions with a certain discriminant. This works over a finite extension of $\mathbb Q_p$, but we'll stick with the base case for simplicity. The following is from sections 3 and 6 of this paper.

Let $j=an+b$, with $0\le b<n$, and assume $j$ satisfies Ore's condition: $${\rm min}({\rm ord}_p(b)n,{\rm ord}_p(n)n)\le j\le {\rm ord}_p(n)n$$ Then the number of totally ramified extensions of $\mathbb Q_p$ of degree $n$ and discriminant $p^{n+j-1}$ is $$\cases{np^{\sum_{i=1}^an/p^i} & b=0\cr n(p-1)p^{\sum_{i=1}^an/p^i+\lfloor (b-1)/p^{a+1}\rfloor}&b>0}$$ Note that if $j$ does not satisfy Ore's condition, there is not a totally ramified extension of degree $n$ and discriminant $p^{n+j-1}$.

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Cool. I'll check that paper. –  pki Nov 6 '11 at 0:54
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