Is x^22 - 3x^11 + 2 Irreducible?

[catcherintherye]

factorize $$x^{22} -3x^{11} + 2$$

right so I have p(x) = (x^11 -2)(x^11 - 1)

(x^11 -2) satifies eisentstein, obviously x-1 is a factor of the second factor. Long division reaps $$x^{10} + x^9 +...+x + 1$$

the solution asserts that this is also irreducible, but I do not see this?? Is this one of those where you substitue x for y+1??

 

[Hurkyl]

Formatting tip: use curly braces {} to group the entire exponent together as one unit.

 

[StatusX]

Yea, any polynomial of the form (x^p-1)/(x-1) can be shown to be irreducible by making the substitution y=x+1 and applying Eisenstein's criteria. You have to do a little work with binomial coefficients, and it's probably easier to prove this general case then the case p=11.