Possible bug, weird base field change when working with polynomial rings

[Henry Talbott]

The following code produces the weird result:

sage: R.<c>=QQ[]
sage: S.<x,y>=R[]
sage: u=FractionField(S)(x^2+y^2)
sage: v = u.numerator()/u.denominator()
sage: print u.numerator().parent()
sage: print v.numerator().parent()

Output:

Multivariate Polynomial Ring in x, y over Univariate Polynomial Ring in c over Rational Field
Multivariate Polynomial Ring in x, y over Fraction Field of Univariate Polynomial Ring in c over Rational Field

Since u.denominator()=1, I expected v to be equal to u, and certainly for their numerators to be over the same base field. I think the base field change may be an issue with the method inverse_of_unit in rings/polynomial/multi_polynomial_element.py.

Any advice would be greatly appreciated! This base field change was causing an error with the dynatomic_polynomial method.

[Travis Scrimshaw]

The general rule is that anytime you do division, it pushes itself out to the ring where it makes sense. You can even see this with integers. The reason for this is because checking if something is a unit or is exactly divisible by the denominator is generally expensive.

Best,

Travis

[Samuel Lelievre]

Workaround: use // instead of / to do division without remainder in the ring.

```

sage: R.<c> = QQ[]

sage: S.<x, y> = R[]

sage: u = FractionField(S)(x^2 + y^2)

sage: v = u.numerator()//u.denominator()

sage: print(u.numerator().parent())

Multivariate Polynomial Ring in x, y over Univariate Polynomial Ring in c over Rational Field

sage: print(v.numerator().parent())

```

[Henry Talbott]

Ok, thank you!

[Ben Hutz]

Actually, I think the issue here is that there isn't actually a division happening, rather, it *is* doing the unit check and inverse of unit calculation within numerator(); but still moves to the fractionfield. When numerator(), on the fractionfield element is called, eventually the function inverse_of_unit() is getting called and that is causing the base ring to change to the fraction field of the base ring. It seems like the whole point of calculating the inverse of a unit in a ring would be to stay within the ring.