Thread: Irreducible Polynomials f(x) = x^7 - x

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!!

Originally Posted by rpietrov

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 contain in any case . The other roots satisfy the equation...

(1)

... or, equivalently, one of the equations...

(2)

(3)

The roots of (2) are , and , the roots of (3) are , and so that is...

(4)

Kind regards

Originally Posted by rpietrov

in Q[x] we think its x(x-1)(x+1)(x^2-x+1)(x^2+x+1)

Right.

in Z/2[x] we think its x(x-1)(x+1)

Try again, it should be .