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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?

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

1 Answer 1

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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

$$\underbrace{1+\dots+1}_{p\text{ 1's in }\mathbb{F}_p[[X]]} = 0,$$

and so

$$\phi\left(\underbrace{1+\dots+1}_{p\text{ 1's in }\mathbb{F}_p[[X]]}\right) = \phi(0) = 0,$$

but

$$\underbrace{\phi(1)+\dots+\phi(1)}_{p\text{ copies of }\phi(1)} = \underbrace{1+\dots+1}_{p\text{ 1's in }\mathbb{Z}_p}=p \neq 0.$$

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