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].