Definition:Modulo Polynomial Division/Divisor

Definition

Let $\map f x$ and $\map g x$ be integral polynomials.

Let $\map f x \div_m \map g x$ denote the operation of polynomial division modulo $m$:

$\map f x \div_m \map g x$ equals the integral polynomial $\map h x$ such that:

$\map g x \times_m \map h x \equiv \map f x \pmod m$

The polynomials $\map g x$ and $\map h x$ are (polynomial) divisors of $\map f x$ modulo $m$.

Examples

Arbitrary Example

Let $\map f x$ be the polynomial:

$\map f x = 2 x^4 - 4 x - 3$

Then $\map f x$ has the following (polynomial) divisors modulo $7$:

$2 x^2 + 3 x + 3$

$x - 2$

$x + 4$

Also known as

A polynomial divisor modulo $m$ is also known as a polynomial factor modulo $m$.

Also see

• Results about polynomial divisors modulo $m$ can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factor modulo $n$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factor modulo $n$