# Definition:Unique Factorization Domain

(Redirected from Definition:Gaussian Domain)

## Definition

Let $\struct {D, +, \circ}$ be an integral domain.

If, for all $x \in D$ such that $x$ is non-zero and not a unit of $D$:

$(1): \quad x$ possesses a complete factorization in $D$
$(2): \quad$ Any two complete factorizations of $x$ in $D$ are equivalent

then $D$ is a unique factorization domain.

## Also known as

A unique factorization domain is also seen as Gaussian domain for Carl Friedrich Gauss.

## Also see

• Results about unique factorization domains can be found here.

## Linguistic Note

The spelling factorization is the US English version.

The UK English spelling is factorisation.