so many questions and not even one line work from you!

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.