Unique Factorization Theorem for Gaussian Integers

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Theorem

The ring of Gaussian integers:

$\struct {\Z \sqbrk i, +, \times}$

forms a unique factorization domain.


That is, every Gaussian integer can be expressed as the product of one or more Gaussian primes, uniquely up to the order in which they appear.


Proof

We have that:

Gaussian Integers form Euclidean Domain
Euclidean Domain is Principal Ideal Domain
Principal Ideal Domain is Unique Factorization Domain.


So $\struct {\Z \sqbrk i, +, \times}$ forms a unique factorization domain.

Hence the result, by definition of unique factorization domain.

$\blacksquare$


Historical Note

The Unique Factorization Theorem for Gaussian Integers was proved by Carl Friedrich Gauss.

This result, and the ideas associated with it, ushered in the field of algebraic number theory.


Sources