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, but the tendency is for the literature to use the factorization form.

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