Unstable minimal polynomial for element of 2-adic Eisenstein Extension Field in pi defined by x^4 - 2*a

[kliem]

The following doctest in src/sage/rings/padics/padic_generic_element.pyx is unstable:

sage: K.<a> = QqCR(2^3,5)
sage: S.<x> = K[]
sage: L.<pi> = K.extension(x^4 - 2*a)
...
sage: elt = L.random_element() 
sage: P = elt.minimal_polynomial()
sage: P(elt) == 0
True

Sometimes it's not exact, ok the ring is not exact.
However, sometimes we get an error:

ArithmeticError: vector is not in free module

and from the documentation it is not clear when and why this happens.

In #29979, a doctest was marked not tested because of this.

Component: algebra

Keywords: minimal polynomial

Issue created by migration from https://trac.sagemath.org/ticket/32111