is_square does not work correctly for Laurent polynomials

[amithazi]

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.

Did you read the documentation and troubleshoot guide?

• I have read the documentation and troubleshoot guide

Environment

- **OS**: Debian 12.0
- **Sage Version**: 10.0

Steps To Reproduce

The method "is_square" for finding square roots of (Laurent) polynomials has a subtle bug.

sage: R.<v> = LaurentPolynomialRing(ZZ)
sage: x = is_square(v^2, root=True)[1]
sage: x, x + v^-1

Expected Behavior

We expect x, x + v^-1 to evaluate to (v, v^-1 + v).

Actual Behavior

Instead, we get (v, 5*v^-1).

Additional Information

There is similar behavior when the coefficients are rational or real.

The problem seems to be that is_square for polynomial rings does not always coerce square roots into the correct parent. For example,

sage: S.<x> = PolynomialRing(ZZ)
sage: is_square(S(1), True)[1].parent()
Integer Ring

These square roots come from factorizations/squarefree decompositions of the polynomial in which the unit is explicitly declared to be a member of the base ring. If the square root of the unit doesn't get multiplied by anything polynomial, the resulting square root remains in the base ring.

[DaveWitteMorris]

Thanks for the bug report, and for your diagnosis of the problem, which was very helpful! A fix is in Pull Request #35876.