I. Let a = 5^(1/3) in R, algebraic over Q with minimal polynomial m(x) = x^3 −5. Write the

following elements of Q(a) as Q-linear combinations of {1, a, a^2}:

(a) u = a^5 + 3a^3 − a + 1

(b) v = (a^2 − 2a + 4)^−1.

My thought is to plug in a and see what u and v equal then.

II. Let F be an extension field of K and let u be a nonzero element of F that is algebraic

over K with minimal polynomial m(x) = x^n + a_(n−1)x^n−1 + · · · + a_1x + a_0. Show that

u^−1 is algebraic over K by finding a polynomial p(x) in K[x] such that p(u^−1) = 0.

Well I know a number u is algebraic if p(u)=0 for a plynomial p(x)

III. Let F be an extension field of K with [F : K] = m < infinity, and let p(x) in K[x] be a

polynomial of degree n that is irreducible over K. Show that if n does not divide m,

then p(x) has no roots in F.

n does not divide m, so we can't have m=nq