# Revision history [back]

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/#narrow/stream/271072-padics/topic/meeting.