## Experimental Mathematics

- Experiment. Math.
- Volume 15, Number 1 (2006), 21-32.

### The Probability That a Random Monic p-adic Polynomial Splits

Joe Buhler, Daniel Goldstein, David Moews, and Joel Rosenberg

#### Abstract

Let {\small $R$} be a complete discrete valuation ring with finite residue field, and let {\small $r_n$} be the probability that a random monic polynomial over {\small $R$} of degree {\small $n$} factors over {\small $R$} into linear factors. We study {\small $r_n$} in detail. Among other things, we show that {\small $r_n$} satisfies an interesting recursion, make a conjecture on the asymptotic behavior of {\small $r_n$} as {\small $n$} goes to infinity, and prove the conjecture in the case that the residue field has two elements.

#### Article information

**Source**

Experiment. Math., Volume 15, Number 1 (2006), 21-32.

**Dates**

First available in Project Euclid: 16 June 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1150476900

**Mathematical Reviews number (MathSciNet)**

MR2229382

**Zentralblatt MATH identifier**

1113.11069

**Subjects**

Primary: 11S05: Polynomials

**Keywords**

p-adic polynomial splits completely

#### Citation

Buhler, Joe; Goldstein, Daniel; Moews, David; Rosenberg, Joel. The Probability That a Random Monic p-adic Polynomial Splits. Experiment. Math. 15 (2006), no. 1, 21--32. https://projecteuclid.org/euclid.em/1150476900