# Integers form Unique Factorization Domain

## Theorem

The integers $\struct {\Z, +, \times}$ form a unique factorization domain.

## Proof

Follows directly from:

$(1) \quad$ the fundamental theorem of arithmetic
$(2) \quad$ the fact that $\struct {\Z, +, \times}$ is an integral domain
$(3) \quad$ the definitions complete factorization and equivalent factorizations.

$\blacksquare$