find the roots of and then find the smallest field which contains the roots and your final answer is
a) find the degree of a minimal splitting field of (x^6 + 1) over Q.
are you sure they don't want you to prove that is an algebraic integer not just an algebraic number, which trivially is if you let n = 1.
b) if a is a any algebraic number prove that there exists a positive integer 'n' such that 'na' is an algebraic number.
is the field F here supposed to be the field of real or complex numbers? in either case noting that and changing w to -w in your identity will solve the problem.
c) if W is a subspace of V and vєV satisfies (v,w) + (w,v) = (w,w) for all wєW prove that (v, w)=0 for all wєW where V is inner product space over F.