How do I proof that x^(p^p) - x is divisible by x^p - x - 1 over Zp, p - any prime number?

[Makuta]

I tried using Freshman's dream theorem and making a^p = a + 1, a^(p^p) = (a+1)^p = a^p + 1, which led to a^p + 1 = a + 2 and it's just not making any sense.

 

[Steven G]

x^p^p is not clear, at least to me.
(2^3)^4 = 2^12 while 2^(3^4) = 2^81

 

[Makuta]

x^p^p is not clear, at least to me.
(2^3)^4 = 2^12 while 2^(3^4) = 2^81

The second option.

 

[blamocur]

Is not $x^{p^p} = x \forall x\in \mathbb Z_p$ by Fermat's little theorem?

 

[Makuta]

Is not $x^{p^p} = x \forall x\in \mathbb Z_p$ by Fermat's little theorem?

I'm not sure, but how is this helpful? I used this theorem and it only makes x≡x (mod p), and for f(x) shows that it has no roots.

 

[blamocur]

I'm not sure, but how is this helpful? I used this theorem and it only makes x≡x (mod p), and for f(x) shows that it has no roots.

It makes $x^{p^p}-x = 0$ in $\mathbb Z_p$, and 0 is divisible by anything, isn't it?

 

[Makuta]

It makes $x^{p^p}-x = 0$ in $\mathbb Z_p$, and 0 is divisible by anything, isn't it?

I guess it is, just seems too simple and thus incorrect. Thank you a lot!

 

[Makuta]

It makes $x^{p^p}-x = 0$ in $\mathbb Z_p$, and 0 is divisible by anything, isn't it?

But again, aren't we saying that every element in Z_p is a root for x^(p^p)-x, but what's the point, if we have the fact that x^p - x - 1 have no roots in Z_p, because if we take a^p - a -1 = 0 in Z_p we get -1 = 0.

 

[blamocur]

But again, aren't we saying that every element in Z_p is a root for x^(p^p)-x, but what's the point, if we have the fact that x^p - x - 1 have no roots in Z_p, because if we take a^p - a -1 = 0 in Z_p we get -1 = 0.

I think misinterpreted your post. I was thinking about divisibility of expressions for any $x\in \mathbb Z_p$. But this would not even require any Fermat theorems since $\mathbb Z_p$ is a field, hence every element is divisible by any other non-zero element.

I suspect you are talking of divisibility of polynomials over $\mathbb Z_p$. This would make a more meaningful problem, but I don't know how to prove it.

 

[Makuta]

but I don't know how to prove it.

Me too, that's why I'm seeking for help...