# Splitting field of $x^{{p}^e}-1$ over $\mathbb Z_p$

I'd like a hint for determining the splitting field of $x^{{p}^e}-1$ over the integers mod $p$, $\mathbb Z_p$, where $e$ is an arbitrary natural number. Thanks.

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– achille hui Jan 28 at 0:47
@achillehui that's right, I think the answer is Zmodp, I mean the splitting field in this case is $Z_p$ itself – User112358 Jan 28 at 1:43

## 1 Answer

Approach 1: Well, there is one obvious root. What is it? What is its multiplicity? Work out the easy stuff, then go from there.

Approach 2: You're asking to solve the equation $x^{p^e} - 1 = 0$....

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Is it $Z_p$ the answer? – User112358 Jan 28 at 1:44
Yep. If you don't guess the factorization, you can easily count the multiplicity of the root $1$ by seeing how many times $x-1$ divides the derivative. Or you can recognize that $1$ is the only $p$-th root of unity in a field of characteristic $p$. – Hurkyl Jan 28 at 3:11
Thank you, I found the factorization indeed, it follows directly from the "freshman's dream". – User112358 Jan 28 at 3:53

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@user59898 $x^{p^e} -1 = (x -1)^{p^e}$ and so the splitting field is just $\Bbb{Z}/p\Bbb{Z}$ itself. – BenjaLim Jan 28 at 10:27
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