Better normalization for fraction field elements - follow up

[mkoeppe]

This is a follow-up on #16268 (and #16993), where the following further improvements were suggested:

• clearing denominators in the numerator and denominator instead of making the leading coefficient of the denominator monic when that makes sense (i.e., for printing, and perhaps for computations in nested rational function fields, but making it fast enough requires some work),
• also normalizing the leading coefficients over non-fields where that makes sense (see also discussion at Broken fraction field of rational polynomial ring #16993).

Related:

• Elements from a Field of Fractions that compare equal should have equal hashes #15297 Elements from a Field of Fractions that compare equal should have equal hashes

CC: @tscrim @cheuberg @Etn40ff @sagetrac-jakobkroeker @bhutz @robertwb @rwst @sagetrac-emassop @JohnCremona @mezzarobba @saraedum

Component: commutative algebra

Issue created by migration from https://trac.sagemath.org/ticket/26339

[yuan-zhou]

comment:2

Unexpected (-1) factor in the numerator and denominator of a sage.rings.fraction_field_element.FractionFieldElement.

SageMath version 9.1.beta3.

sage: P.<a,b,c>=QQ[]
sage: f = 1/a/(b-a)/(c-a)+1/b/(a-b)/(c-b)+1/c/(a-c)/(b-c); f
(-1)/(-a*b*c)
sage: f.numerator()
-1
sage: f.denominator()
-a*b*c

[mkoeppe]

comment:4

pushing these forward to 9.2

[mkoeppe]

comment:6

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.