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Q: Let $p$ be a prime and let $e \ge 2$. The quotient ring $\mathbb{Z}/p^e\mathbb{Z}$ and the finite field $\mathbb{F}_{p^e}$ are both rings and have the same number of elements. Describe some ways in which they are intrinsically different.

My textbook confirms my understanding that $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{F}_{p}$? are the same thing.

With $\mathbb{Z}/p^e\mathbb{Z}$, there will be elements that don't have multiplicative inverses, so it is not a field. What is $\mathbb{F}_{p^e}$ if it's not the same as $\mathbb{Z}/p^e\mathbb{Z}$?

clay
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  • $Z/p^e$ has zero divisors for $e>1$, $\mathbb{F}_{p^e}$ does not have zero divisors. See also here for the difference of $\mathbb{Z}/p$ and $\mathbb{F}_p$. – Dietrich Burde Oct 05 '16 at 16:59
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    $\mathbb{F}{p^e}$ is the field with $p^e$ elements, and it is not $\mathbb{Z}/p^e\mathbb{Z}$, since the latter is not a field :-) For a construction of $\mathbb{F}{p^e}$, you may read the notes of Keith Conrad: http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/finitefields.pdf –  Oct 05 '16 at 17:30

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