Math 120B Name: Spring 2021 Take-Home Midterm Exam Student ID: Due Mon 5/12/2020 at 11:59pm in Canvas • There are 6 questions for a total of 100 points

Math 120B Name: Spring 2021 Take-Home Midterm Exam Student ID: Due Mon 5/12/2020 at 11:59pm in Canvas • There are 6 questions for a total of 100 points. • Some questions have several parts. • For full credit show ALL of your work, explain your process fully. Make sure that I can understand exactly HOW you got your answer. • Please box your final answer, when applicable. Question: 1 2 3 4 5 6 Total Points: 18 16 20 15 21 10 100 Score: Page 1 of 2 1. (a) (8 points) Find the remainder of 51011 modulo 303 (b) (10 points) Find all x ∈ Z solutions to 155x ≡ 75 mod 65, if any exist. 2. Consider the following problems in Z7 [x]: (a) (6 points) Find all of the roots of the polynomial h(x) = x3 + 4x2 + x + 1 ∈ Z7 [x] (b) (10 points) For f (x) = x6 + 3x5 + 4x2 − 3x + 2 and g(x) = 3x2 + 2x − 3 in Z7 [x], find q(x) and r(x) as described by the division algorithm so that f (x) = g(x)q(x) + r(x). Be sure to reduce your final answers mod 7. 3. Determine whether the following polynomials are irreducible in Z[x] (Hint: you should be able to prove these using the methods from lecture in section 23). (a) (10 points) f (x) = x3 − 82x + 432 (b) (10 points) g(x) = x6 − 1 = x5 + x4 + x3 + x2 + x + 1 x−1 4. (15 points) Count the number of irreducible polynomials of degree 3 in the polynomial ring Z5 [x] (Hint: you do not have to list every polynomial to make your argument formal). 5. Determine whether the following statements are true or false. Justify with a proof or a counterexample. (a) (7 points) φ : R × R −→ C with φ((a, b)) = a + bi is an isomorphism. (b) (7 points) 3Z/9Z ∼ = Z3 as rings. (c) (7 points) For a ring R, it is possible to have a, b 6∈ R× and ab ∈ R× 6. (10 points) Give an example of a non-commutative ring of characteristic 2, or prove that none exists.