Unique Factorization Theorem for Gaussian Integers

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Theorem

For every Gaussian integer $n$ such that $\left\vert{n}\right\vert > 1$, $n$ can be expressed as the product of one or more Gaussian primes, uniquely up to the order in which they appear.


Proof


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.