# Counting Separable Polynomials in $\mathbb{Z}/n[x]$

## TL;DR

This paper derives formulas to count separable polynomials over the ring rac{n}{x}, extending previous results, and provides explicit counts based on the prime factorization of n.

## Contribution

It introduces explicit formulas for counting separable polynomials over rac{n}{x}, generalizing earlier work by Carlitz.

## Key findings

- Number of separable polynomials is rac{(n)n^d}{(n)} rac{(n)n^d}{(n)} rac{(n)n^d}{(n)}
- Formulas depend on the prime factorization of n and Euler's totient function.
- Extends previous results to more general rings and polynomial degrees.

## Abstract

For a commutative ring $R$, a polynomial $f\in R[x]$ is called separable if $R[x]/f$ is a separable $R$-algebra. We derive formulae for the number of separable polynomials when $R = \mathbb{Z}/n$, extending a result of L. Carlitz. For instance, we show that the number of separable polynomials in $\mathbb{Z}/n[x]$ that are separable is $\phi(n)n^d\prod_i(1-p_i^{-d})$ where $n = \prod p_i^{k_i}$ is the prime factorisation of $n$ and $\phi$ is Euler's totient function.