Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
Hint: Freshman's dream –  achille hui Jan 28 '13 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 '13 at 1:43

1 Answer 1

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 $x^{p^e} - 1 = 0$....

share|improve this answer
    
Is it $Z_p$ the answer? –  User112358 Jan 28 '13 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 '13 at 3:11
    
Thank you, I found the factorization indeed, it follows directly from the "freshman's dream". –  User112358 Jan 28 '13 at 3:53
1  
@user59898 $x^{p^e} -1 = (x -1)^{p^e}$ and so the splitting field is just $\Bbb{Z}/p\Bbb{Z}$ itself. –  user38268 Jan 28 '13 at 10:27

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.