Is $\mathbb{Z}_p \cong \mathbb{Z}[[X]]/(X-p)$?

Question

Let $p$ be a prime number and denote by $\mathbb{Z}_p$ the ring of p-adic integers. Denote by $\mathbb{Z}[[X]]$ the ring of formal power series with integer coefficients and let $\frac{\mathbb{Z}[[X]]}{ (X-p)}$ denote the quotient of $\mathbb{Z}[[X]]$ by the ideal $(X-p)$. Does there exist a ring isomorphism $\mathbb{Z}_p \cong \frac{\mathbb{Z}[[X]]}{ (X-p)}$ ? If the answer is yes, a constructive proof is preferred.

Answer 1

Indeed, there is an isomorphism of topological rings $\mathbb{Z}[[X]]/\langle X - p\rangle \cong \mathbb{Z}_p$.

Hint. Construct a continuous surjective homomorphism $\varphi : \mathbb{Z}[[X]] \to \mathbb{Z}_p$ with $\ker(\varphi)=\langle X - p \rangle$.

The information $\varphi(X-p)=0$ already tells you what $\varphi(X)$ must be, and then extend the definition to all of $\mathbb{Z}[[X]]$.