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Mathematics 2215: Rings, fields and modules Homework exercise sheet 5 Due 4.50pm, Thursday 16th December 1. Suppose that α ∈ C with α3 − α2 + α + 2 = 0. Write α(1 + α)3 and 1 in the form a + bα + cα2 where a, b, c ∈ Q. 1−α 2. (a) Find an irreducible element of Z3 [x] with degree 2, and use this to construct a field F with 9 elements. (b) For your field F , find an element a ∈ F so that a8 = 1F and ak 6= 1 for 1 ≤ k ≤ 7. Deduce that F × is a cyclic group under multiplication. 3. Let L be a field, and let F be a subfield of L. If α, β ∈ L and K = F (α), explain why F (α, β) = K(β). √ √ 4. Compute [Q( 3, 5) : Q]. 5. Show that if K : F (that is, if K and F are fields, and K is a field extension of F ) and [K : F ] is prime, then K = F (α) for some α ∈ K which is algebraic over F . 6. If K : F and α ∈ K is algebraic over F and the degree of the minimum polynomial of α over F is odd, show that F (α) = F (α2 ). 7. Recall from complex analysis that if θ ∈ R, then eiθ = x + iy where x = cos(θ) and y = sin(θ). Let α = eπi/6 . (a) What is the minimum polynomial of α over C? (b) Show that R(α) = C and find the minimum polynomial of α over R. (c) Show that [Q(α) : Q] = 4 and find the minimum polynomial of α over Q. 8. (a) Factorise the polynomial x12 − 1 into monic irreducible elements of C[x]. Hint: what is (eπi/6 )12 ? (b) [Optional] Repeat this exercise for R[x] and for Q[x].