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GaussianIntegers
GaussianIntegers is an option for FactorInteger, PrimeQ, Factor and related functions which specifies whether factorization should be done over Gaussian integers.
With GaussianIntegers
->
False, factorization is done over the ordinary ring of integers  .
With GaussianIntegers
->
True, factorization is done over the ring of integers with  adjoined  .
Example: FactorInteger[13, GaussianIntegers -> True] .
The Gaussian primes used when GaussianIntegers
->
True are chosen to have both real and imaginary parts positive.
The first entry in the list given by FactorInteger with GaussianIntegers
->
True may be -1 or -I.
See the Mathematica book: Section 3.2.4.
See also: Extension, ComplexExpand.
Further Examples
Although  is prime over the integers, it factors nontrivially over the complex integers and so is not a prime in that ring.
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See the Further Examples for Factor, DivisorSigma.
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