As already mentioned in the introduction of this chapter, the domain of Gaussian rationals is a field. Therefore all field functions are applicable to this domain and its elements (see chapter Fields). This section gives further comments on the definitions and implementations of those functions for the the Gaussian rationals. All functions not mentioned here are not treated specially, i.e., they are implemented by the default function mentioned in the respective section.
Conjugates
The field of Gaussian rationals is an extension of degree 2 of the
rationals, its prime field. Therefore there is one further conjugate of
every element a + b*E(4), namely a - b*E(4).
Norm, Trace
According to the definition of conjugates above, the norm of a Gaussian
rational a + b*E(4) is a^2 + b^2 and the trace is
2*a.
GAP 3.4.4