Let q be a root of p(x) = x^3 + x^2 + 1 in an extention field of Z2 (integers modulus 2). Show that Z2(q) is a splitting field of p(x by finding the other roots of p(x)
hint: this question can be greatly simplified by using the frobenius automorphism to find these zero's
frobenius automorphism : f(a) = a^p
The Attempt at a Solution
So we are given one root and need to prove that this is enough to when adjoined to the original field to get the other two roots...
so p(q) = 0
and p(x) = (x-q)m(x) where m(x) is of degree 2
I'm a little confused on how to use the frobenius automorphism here or even why it would help at all...
anyone have any insight for me?