CoCoA » CoCoALib

What should be the internal representation of an element of QQ[i]?

1. a pair of BigRat values (real and imag parts)
2. a triple of BigInt values (real, imag and common denom)
3. a 4-tuple of BigInt values (real & imag of numerator, real & imag of denominator)

Representation (3) is unnormalized, i.e. the same value has many different representations; division is very simple; addition and multiplication are more costly than in repr (2).

Repr (2) is the most "specialized"; I would expect it to be faster at run-time than the other two reprs.

Repr (1) is perhaps the most natural, but having two separate denominators is likely to be a disadvantage at run-time (more memory space, and slower computation than repr (1))

Recalling that every ring (except RingZZ) in CoCoALib has a BaseRing and an "extension type". What should the BaseRing and "extension type" of QQ[i] be?

Ideally it should behave as if it were a quotient of a polynomial extension. But if we do this then it must be possible to obtain a "representative" being a univariate polynomial in the polynomial ring QQ[i].

Perhaps there should be a special "simple algebraic extension" type whose BaseRing is the coeff ring, and the intermediate polynomial ring is not accessible?