For anyone who's ever heard of roots of unity or cyclotomic polynomials this is very simple, so let's assume we haven't.
We will only use one not completely elementary tool and that is the fact that an n-th degree polynomial has n complex roots (counting multiplicities).
Substituting a for c for b in the given system of equations yields an order 8 polynomial in a, therefore there are 8 solutions (the exact number is not important; all we will use is the fact that there are more than zero and fewer than infinity solutions other than zero and one). Exactly two of those are ruled out because of a=b, viz. a=1 and a=0. We can therefore determine the sum of all nonzero solutions and then subtract 1.
Observe that if a and a' are solutions other than zero then so are 1/a and aa'. From this it follows that if we take one solution a and form the products aa1,aa2,... for all nonzero solutions ai then each solution is replaced with a different solution but the set as a whole is unchanged. It follows that the sum S must satisfy aS=S for all solutions a which is only possible if S=0; subtracting 1 we get the final answer -1.