left_divides for Ore polynomials

[kryzar]

Steps To Reproduce

sage: Fq = GF(2)
sage: A.<T> = Fq[]
sage: Ktau = phi.ore_polring()
sage: t = phi.ore_variable()
sage: f = (z + 1)*t^10 + t^9 + t^8 + z*t^6 + (z + 1)*t^5 + (z + 1)*t^4 + t^3 + z*t^2 + t
sage: f.left_divides(f)
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
...
NotImplementedError: inversion of the twisting morphism Frobenius endomorphism x |--> x^2 of Finite Field in z of size 2^2 over its base
sage: 

Expected Behavior

f.left_divides(f) should return True

Actual Behavior

An exception is raised.

Additional Information

I appreciate it probably has to do with computing $q^{\mathrm{something}}$-th roots of coefficients in the base field, which may not exist. Wouldn't there be a way to return False as long as one coefficient doesn't have any of its relevant roots in the base field?

Environment

• OS: Linux
• Sage Version: SageMath version 10.5, Release Date: 2024-12-04

Checklist

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