Definition:Division over Euclidean Domain/Examples

Examples of Division over Euclidean Domain

Let $a, b \in \Z$ be integers such that $b \ne 0$.

$\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$

where $q$ is the quotient and $r$ is the remainder.

The process of finding $q$ and $r$ is known as (integer) division of $a$ by $b$, and we write:

$a \div b = q \rem r$

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.

Let $\map A x$ and $\map B x$ be polynomials in $F \sqbrk X$ such that the degree of $B$ is non-zero.

$\exists \map Q x, \map R x \in F \sqbrk X: \map A x = \map Q x \map B x + \map R x$

such that:

$0 \le \map \deg R < \map \deg B$

where $\deg$ denotes the degree of a polynomial.

The process of finding $\map Q x$ and $\map R x$ is known as polynomial division of $\map A x$ by $\map B x$, and we write:

$\map A x \div \map B x = \map Q x \rem \map R x$