GaussianIntegers

GaussianIntegers
is an option for FactorInteger, PrimeQ, Factor, and related functions that 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 i adjoined .

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 or I.

Factor a polynomial over

Factor an integer over

Factor a polynomial over

Out[1]=

Factor an integer over

Out[2]=

By default polynomial factorization is performed over the rationals:

This specifies that the factorization should be done over

By default integer factorization is performed over the integers:

This specifies that the factorization should be done over the Gaussian integers:

A number prime over the integers may not be prime over the Gaussian integers: