Power Series Ring over p-adics: TypeError: unhashable

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(Not a solution). The problem comes from the fact that sage is not able to derive enough information on Zpi

sage: ZZq.<xi> = Qq(3^1)
sage: R.<x_pi> = ZZq[]
sage: Zpi.<pi> = QuotientRing(R, R.ideal(x_pi^(p-1)+p))
sage: print(Zpi.is_field())
True
sage: ZZq.<xi> = Qq(3^2)
sage: R.<x_pi> = ZZq[]
sage: Zpi.<pi> = QuotientRing(R, R.ideal(x_pi^(p-1)+p))
sage: print(Zpi.is_field())
Traceback (most recent call last):
...
NotImplementedError

In this case there is a specialized type for this p-adic base field which is called a "relative extension" in SageMath, i.e., an Eisenstein extension put on top of an unramified extension:

sage: K.<u> = Qq(3^2)
sage: R.<x> = K[]
sage: L.<pi> = R.extension(x^2 - 3)
sage: R.<x> =  L[[]]
sage: R
Power Series Ring in x over 3-adic Eisenstein Extension Field in w defined by x^2 + 3 over its base field

When you used QuotientRing, it only gives you the formal quotient but does not understand enough about the underlying structure, e.g., that this is a field (which I suspect is because the base ring is not exact.) In any case, the above construction might also not prove too useful. It's tricky to get the interplay between the non-exact p-adic base and the non-exact power series ring right and SageMath is (last time I checked) not particularly good about it; but it depends a lot on your application. I don't know what's your application, but sometimes one can replace the base ring by an exact ring, e.g., a Henselization or some other number field based approach.

PS: If you want to discuss your application, some of the people who care about p-adics in SageMath are meeting most Thursdays at 11pm CET at https://sagemath.zulipchat.com/#narro....

Depending on what you want to do, you can maybe also consider using the constructor TateAlgebra (it is not exactly the same than power series but much effort is done to handle precision correctly):

sage: K.<u> = Qq(3^2)
sage: L.<pi> = K.extension(x^2 - 3)
sage: L                                                                         
3-adic Eisenstein Extension Field in pi defined by x^2 - 3 over its base field
sage: A.<x> = TateAlgebra(L)
sage: (1 - pi*x).inverse_of_unit()
(1 + O(pi^40)) + (pi + O(pi^40))*x + ... + O(pi^40 * <x>)