Density of $p$-adic polynomials generating extensions with fixed splitting type

John Yin

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arXiv · 2211.10425

Density of $p$-adic polynomials generating extensions with fixed splitting type

John Yin

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Abstract

We prove that the density of polynomials $P(x)=\sum_{i=0}^n a_n x^n$ over a local field $K$ generating an \'etale extension with specified splitting type is a rational function in terms of the size of the residue field of $K$ in the case where the splitting type is tame. Moreover, we give a computable recursive formula for these densities and compute the asymptotics of this density as the size of the residue field tends to infinity.

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John Yin. 2022-11-18. Density of $p$-adic polynomials generating extensions with fixed splitting type. https://arxiv.org/abs/2211.10425

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