units in polynomial rings with prime power characteristic

[tdupu]

sage: p = 101
sage: S.<t> = PolynomialRing(ZZ.quotient_ring(p^3))
sage: f = 1+p*t^2
sage: f.is_unit()
True

sage: S.<t> = PolynomialRing(ZZ.quotient_ring(p^3),1)
sage: t = S.0
sage: f = 1+p*t^2
sage: f.is_unit()
False
sage: f.inverse_of_unit()
...
ArithmeticError: Element is not a unit.

sage: S1.<t> = PolynomialRing(ZZ.quotient_ring(p^3),1)
sage: S2.<tt> = PolynomialRing(ZZ.quotient_ring(p^3))
sage: S1
Multivariate Polynomial Ring in t over Ring of integers modulo 1030301
sage: S2
Univariate Polynomial Ring in tt over Ring of integers modulo 1030301
sage: S1 == S2
False

Depends on #22454

Component: algebra

Stopgaps: todo

Author: Mark Saaltink

Branch/Commit: u/vdelecroix/11537 @ 718b0e2

Issue created by migration from https://trac.sagemath.org/ticket/11537

[tdupu]

Attachment: units_mod_p.sws.gz

[tdupu]

comment:1

--There are lots of "inverse_of_unit" methods one should implement while they are fixing this the attached notebook shows one way this can be done from prime power characteristic.