Definition:Unique Factorization 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.

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62$. Factorization in an integral domain
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Gaussian domain or unique factorization domain