Integers form Unique Factorization Domain
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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.
$\blacksquare$