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Exercise: Let $p$ be a prime number. Then, the polynomial \begin{equation} \frac{X^{p^n}-1}{X^{p^{n-1}}-1} \end{equation} is irreducible over $\mathbb Z[X]$, for any integer $n \geq 1$.

I'm able to work out the case $n=1$ (substitution $Y=X+1$ and Eisenstein's criterion); how can I prove the general case?

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  • $\begingroup$ The same argument that works for $n=1$ can work for any $n$, with a bit more work. $\endgroup$ – Thomas Andrews Mar 17 '14 at 16:04
  • $\begingroup$ Could you give me some more hints? $\endgroup$ – Francesco Genovese Mar 17 '14 at 16:36
  • $\begingroup$ How did you prove it for $n=1$? $\endgroup$ – Thomas Andrews Mar 17 '14 at 16:56
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Interesting fact: This is the $p^n$th cyclotomic polynomial $\Phi_{p^n}(x)$. Proving this for any $n$ is similar to the case $n=1$.Assume we are working over a field $\Bbb F$. We have $(x+1)^{p^{n-1}}=x^{p^{n-1}}+1\mod p$ . This allows us to write in $\Bbb F_p[x]$ $$\begin{align}\frac{(x+1)^{p^{n}}-1}{(x+1)^{p^{n-1}}-1}\end{align}=\\((x+1)^{p^{n-1}})^{p-1}+((x+1)^{p^{n-1}})^{p-2}+\dots+((x+1)^{p^{n-1}})+1=\\(x^{p^{n-1}}+1)^{p-1}+(x^{p^{n-1}}+1)^{p-2}+\dots+(x^{p^{n-1}}+1)+1=\\ \frac{(x^{p^{n-1}}+1)^{p}-1}{(x^{p^{n-1}}+1)-1}=x^{(p-1)p^{n-1}}$$

Which shows that $p$ divides all the coefficients but that of the highest degree. Note that the constant term is $p$(Prove it) which is not divisible by $p^2$, and you get your result by the Eisentstein criterion.

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