(13) Find the degree of each of the following fields Q(√5 + i), Q(√5), Q(i), Q(√5, i), Q(i√5) over Q. Decide which of the above fields is contained in which (you may draw a diagram).
(14) Write down all the complex roots of the polynomial f(x) = x⁵ - 3. Let α = fifth root of 3 be the real root of f and let β be any of the non-real roots. Find [Q(α) : Q) and (Q(α, β) : Q(α)]. What does this imply about [Q(α, β) : Q)? Prove that all the complex roots of f belong to Q(α, β). The smallest subfield of C which contains all the roots of an irreducible polynomial is called the splitting field of that polynomial. Let C₅ denote a fifth root of unity. Prove that C₅ ∈ Q(α, β). Is it true that C₅ ∈ Q(β)?
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