If we adjoin a square root of -1, usually denoted by i, to the field of rationals we obtain a field that is an extension of degree 2. This field is called the Gaussian rationals and its ring of integers is called the Gaussian integers, because C.F. Gauss was the first to study them.
In GAP Gaussian rationals are written in the form
a + b*E(4),
where a and b are rationals, because
E(4) is GAP's name for
i. Because 1 and i form an integral base the Gaussian integers are
written in the form
a + b*E(4), where a and b are integers.
The first sections in this chapter describe the operations applicable to Operations for Gaussians).
The next sections describe the functions that test whether an object is a Gaussian rational or integer (see IsGaussRat and IsGaussInt).
The GAP object
GaussianRationals is the field domain of all Gaussian
rationals, and the object
GaussianIntegers is the ring domain of all
Gaussian integers. All set theoretic functions are applicable to those
two domains (see chapter Domains and Set Functions for Gaussians).
The Gaussian rationals form a field so all field functions, e.g.,
are applicable to the domain
GaussianRationals and its elements (see
chapter Fields and Field Functions for Gaussian Rationals).
The Gaussian integers form a Euclidean ring so all ring functions, e.g.,
Factors, are applicable to
GaussianIntegers and its elements (see
chapter Rings, Ring Functions for Gaussian Integers, and
The field of Gaussian rationals is just a special case of cyclotomic fields, so everything that applies to those fields also applies to it (see chapters Cyclotomics and Subfields of Cyclotomic Fields).
All functions are in the library file