# Is it true that $\mathbb{Z}_p\cong \mathbb{F}_p[[X]]$? [closed]

Let $$f(X)=\sum_{n=0}^{\infty}a_n X^n \in \mathbb{F}_p[[X]]$$.

Define $$\phi:\mathbb{F}_p[[X]]\to \mathbb{Z}_p, \; \; \;\phi(f(X)):=\sum_{n=0}^{\infty}a_n p^n.$$

I think that this is an isomorphism by $$p$$-adic expansion and uniqueness.

This is correct? If not, why is it wrong?

• That is not correct as explained by sbares below. We have $\mathbb{Z}_p \cong \mathbb{Z}[[X]]/(X-p)$ though.
– Con
Aug 1, 2021 at 9:31

There are multiple straightforward ways to see that $$\mathbb{F}_p[[X]]$$ and $$\mathbb{Z}_p$$ are not isomorphic; eg. one has characteristic $$p$$, the other has characteristic $$0$$. In fact, we can use the characteristic to find a concrete problem with your "isomorphism" $$\phi$$: In $$\mathbb{F}_p[[X]]$$ we have
$$\underbrace{1+\dots+1}_{p\text{ 1's in }\mathbb{F}_p[[X]]} = 0,$$
$$\phi\left(\underbrace{1+\dots+1}_{p\text{ 1's in }\mathbb{F}_p[[X]]}\right) = \phi(0) = 0,$$
$$\underbrace{\phi(1)+\dots+\phi(1)}_{p\text{ copies of }\phi(1)} = \underbrace{1+\dots+1}_{p\text{ 1's in }\mathbb{Z}_p}=p \neq 0.$$