Guys I have a couple of questions. All help is appreciated.

1. Is X^3-3 an irreducible polynomial over the real numbers R? over the rational numbers Q? Prove your assertions.

I know the cube root of 3 is a real number so its not irreducible over the reals and it is for the rationals since the cube root of 3 is not a rational number but is that the proof?

2. Q the field of rationals, is

A. Q(sqrt(2)) = {a + b(sqrt(2)) | a,b in Q } a subfield of the real numbers?

B. Q(cube root(3)) = {a + b(cube root(3)) + c(cube root (9)) | a,b,c in Q} a field?

I need some serious help for those two above and this one below.

3. V_R is the set of all differentiable real valued functions of a real variable, is {1, x, x^2,...} a linearly independent subset of V_R? is it a basis of V_R? Prove your assertions.

thanks for the help