The Proportion of Irreducible p-adic Polynomials

TL;DR

This work determines the exact local density of irreducible monic $p$-adic polynomials of degree $n$ in $ obreak \mathbb{Z}_p[x]$ under Haar measure. The authors employ elementary techniques—Hensel's Lemma, Newton polygons, Gauss' Lemma, and a shift-invariance argument—to lift counting data from reductions mod $p$ to $p^k$ and, crucially, to show irreducibles are equidistributed across the $x^{n-1}$ term, reducing to the no-$x^{n-1}$-term slice. They prove an asymptotic density $I = \frac{1}{n} + O(p^{-n/2})$ and derive exact rational formulas for two cases: (i) prime degree $n=r$ with $(r,p)=1$, and (ii) quartics ($n=4$) with $p \neq 2$. The resulting closed forms—$I = \dfrac{(p^r-p)(1-p^{-(r+1)/2}) + r(p-1)(1 - p^{-(r+1)(r-1)/2})}{r p^r (1-p^{-r(r+1)/2 + 1})(1-p^{-(r+1)/2})}$ for primes, and a specific quartic formula—highlight that these irreducibility proportions are rational functions in $p$, and extend the landscape of exact local densities beyond prior root-count results. The findings have potential implications for local-global analyses and splitting-type distributions of $p$-adic polynomials.

Abstract

We attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational functions in $p$. This relates to previous work done to find exact proportions of $p$-adic polynomials of degree $n$ which have $k$ roots.

Theorems & Definitions (29)

• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof