# Formal power series rings and p-adic solenoids

For any prime $$p$$, the ring of $$p$$-adic integers can be generated as the quotient of the formal power series ring $$\Bbb Z[[x]]/(x-p)$$. My questions:

• If we instead use $$\Bbb Q[[x]]/(x-p)$$, do we get the field of $$p$$-adic numbers?
• If we instead use $$\Bbb R[[x]]/(x-p)$$, do we get the $$p$$-adic solenoid?
• Is there any sensible interpretation of $$\Bbb C[[x]]/(x-p)$$?

Very generally, if $$R$$ is a commutative ring and $$f\in R[[x]]$$, then $$f$$ is a unit iff the constant term of $$f$$ is a unit in $$R$$ (if the constant term is a unit, then you can build the coefficients of an inverse to $$f$$ one-by-one). So if $$p$$ is a unit in $$R$$, then $$x-p$$ is a unit in $$R[[x]]$$, and $$R[[x]]/(x-p)$$ is the zero ring.
Note that if you instead took $$\mathbb{Z}[[x]]/(x-p)$$ and formally inverted $$p$$, then you would get the $$p$$-adic rationals. The difference between that and $$\mathbb{Q}[[x]]/(x-p)$$ is that in $$\mathbb{Q}[[x]]$$ you can have a power series which has coefficients with unbounded powers of $$p$$ in the denominators. Such a power series cannot be written as a power series with coefficients in $$\mathbb{Z}$$ divided by any fixed power of $$p$$. This is in fact exactly what happens with the inverse of $$x-p$$.