in Q[x], Z/2[x], and Z/7[x].
in Q[x] we think its x(x-1)(x+1)(x^2-x+1)(x^2+x+1)
in Z/2[x] we think its x(x-1)(x+1)
and we have no idea about Z/7[x] we've tried using Little Fermats Theorem.
THANKEEE YOU!!
The polynomial $\displaystyle P(x)= x^{7}-x$ contain in any case $\displaystyle x$. The other roots satisfy the equation...
$\displaystyle x^{6} \equiv 1\ , \text{mod}\ 7$ (1)
... or, equivalently, one of the equations...
$\displaystyle x^{3} \equiv 1\ , \text{mod}\ 7$ (2)
$\displaystyle x^{3} \equiv -1\ , \text{mod}\ 7$ (3)
The roots of (2) are $\displaystyle 1$, $\displaystyle 2$ and $\displaystyle 4$, the roots of (3) are $\displaystyle -1$, $\displaystyle -2$ and $\displaystyle -4$ so that is...
$\displaystyle x^{7}-x= x\ (x+1)\ (x-1)\ (x+2)\ (x-2)\ (x+4)\ (x-4)$ (4)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$