^{1/3}+ 2

^{1/2}.

I would just use maple but I do not have it installed on this machine.

I found the polynomial and verified that this is indeed a root. I only have Eisenstiens criterion for determining whether it is irreducible, and I can not apply it in this case. Do you have another method? I have not tried substituting x=x+1 or x=x-1 or other substitutions.

The polynomial is x

^{6}-6x

^{4}-4x

^{3}+12x

^{2}-24x-4