(a) Prove that the polynomial g(x) = x^{2}- sqrt (2) is irreducible over Q(sqrt(2)). It follows that g is the minimal polynomial for^{4}(sqrt(2)) <--- (the fourth root of the square root of two)

Hint: What are the elements of Q(sqrt(2))?

(b) Use part (a) to show that the elements of Q(^{4}sqrt(2)) are of the form a + b^{4}(sqrt(2)) + c sqrt(2) + d^{4}(8) for a,b,c,d in Q, uniquely.

(c) Find the minimal polynomial for^{4}(sqrt (2)) over Q, and show that it is minimal.