# Contents

## Definition

Given a field $K$ and a polynomial $p \in K[x]$, a polynomial modulo $p$ is an element of the quotient ring $q \in K[x]/p K[x]$. The quotient ring itself is called the ring of polynomials modulo $p$.

## Examples

• The Gaussian rationals are the rational polynomials $\mathbb{Q}[x]$ modulo $x^2 + 1$.

• The complex numbers are the real polynomials $\mathbb{R}[x]$ modulo $x^2 + 1$.

• The dual numbers are the real polynomials $\mathbb{R}[x]$ modulo $x^2$.

## Properties

If $p$ is an irreducible element in $K[x]$, then the quotient ring $K[x]/p K[x]$ is a field.

If $p^n$ is a power of an irreducible element $p \in K[x]$, then the quotient ring $K[x]/p^n K[x]$ is a local ring.

Given an irreducible element $p \in K[x]$, one could construct the completion of a ring

$K[x]_p \coloneqq \underset{n \to \infty}\mathrm{colim} K[x]/p^n K[x]$

which is a discrete valuation ring, a local integral domain with a Dedekind-Hasse norm. In particular, if $p = x - a$ is a monic polynomial of degree one with $a \in K$ a constant, then $K[x]_{x - a} \coloneqq K[[x - a]]$ is the ring of formal power series expanded around $a$.

If $p$ is a square-free element of $K[x]$, then the quotient ring $K[x]/p K[x]$ is a reduced ring.

In general, the quotient ring $K[x]/p K[x]$ is a prefield ring, whose monoid of regular elements are the equivalence class of polynomials $q$ modulo $p$ such that the greatest common divisor is in the group of units of the polynomial ring.

$\gcd(p, q) \in K[x]^\times$