MA2ALA Coursework

This Assessment should be submitted online via BLACKBOARD (via the submission point in the
Assessments folder) by noon on 23rd April. Please write your answers, showing full working,
on paper and then upload a scan of your work in the form of a single PDF file. The marks
available for each question are shown on the right hand side of the page. The maximum mark
possible is 40. The assessment will count for 10% of your final MA2ALA mark.

1. (a) Classify all non-zero elements of Z15 into units and zero divisors.
   4 marks
   (b) Pick any unit [u] that is not [1] or [14] and find n such that [u]
   n = [1].
   2 marks
   (c) For the element chosen in part (b), find the inverse.
   1 mark
   (d) Does the equation [5]x = [0] have any non-zero solutions? If no, why not? If yes, list
   all solutions.
   3 marks
2. Use Noether’s Isomorphism Theorem to show Z30/(5) ∼= Z5.
   10 marks
3. (a) For each pair of polynomials f, g ∈ Q[x], find the greatest common divisor.
   (i) f = x
   5 + 2x
   4 − x
   2 + 1, g = x
   4 + 1;
   (ii) f = 2x
   5 + 2x
   3
   , g = 2x
   3 − 2x
   2 + 2x − 2.
   4 marks
   (b) For each pair of polynomials in part (a), find a polynomial p(x) ∈ F[x] such that
   (f, g) = (p).
   2 marks
   (c) For each pair of polynomials in part (a), decide if the ideal (f, g) is prime. Justify
   your answer fully.
   1
   4 marks
4. (a) Compute the minimal polynomial of α =
   √
   3 + √
   −2 over Q. Justify your answer.
   6 marks
   (b) Consider the polynomial,
   f(x) = x
   35 − 12x
   26 − 9x
   24 + 39x
   16 + 21x
   11 − 27x
   4 + 3x + 6.
   Let β denote one of the roots of f(x) in C, and let K = Q(β). Give a basis for K as
   a vector space over Q. Justify your answer.
   4 marks
   2