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According to the title, the equation is :

$$x^{2p}- x^p=[6]$$

for $p$ prime, in $\mathbb {Z}_p$.

It's known that $a^p \equiv a \mod p$. So, if the equation was $a^p -a=[6]$ , $a=6$.

I tried to make some manipulations using as tool this theorem, but I am lost here.

I can't figure this out after lots of time thinking, anything else could I do?

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$x^{2p} = (x^p)^2$. So, what equation do you get? – Daniel Fischer Jul 2 '13 at 23:44
up vote 4 down vote accepted

As you know $x^p \equiv x \pmod{p}$, the equation you want to solve is equivalent to:

$$x^2 - x \equiv 6 \pmod{p}$$

The quadratic polynomial factors: $x^2 -x -6 = (x+2)(x-3)$. So does that get you there?

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Thanks a lot :D ! – HipsterMathematician Jul 2 '13 at 23:57

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