Solvability of Artin-Schreier Polynomial I'm having a hard time trying to prove that the polynomial
f(x) = x^p - x - 1 in Z_p[x]
is not solvable by radicals even though its Galois Group is solvable.
So far, I have shown that the polynomial is irreducible in the base field and that its Galois group is isomorphic to Z_p. Since the Galois group is cyclic, it is solvable. I have also shown that if n is a root of f then the pth root of n is not a root of f by the definition of solvable by radicals given to us, which is definition 2 in the link below:
http://www.math.brown.edu/~abrmovic/MA/f1314/251/Zijian-notes.pdf

Show that the polynomial is irreducible. Show that the decomposition field of that polynomial is generated by any one of its roots; in particular, the degree of the decomposition field is then $p$. Show that thay field is not generated by the pth roots of any element of the prime field —or that if an element of that field is a pth root of the prime field, it is already in the prime field.
