Mathematics 351 – Abstract Algebra I Midterm Exam 2 – November 14, 2007

Directions: Place all work you want to have considered for credit in the blue exam book. You have until 9:00pm to work on the exam if you need that much time. There are 100 possible points. Some partial credit will be given for relevant definitions, statements of theorems, etc. Even if you cannot see how to complete some part, don’t leave it blank. I. Terminology. A) (5) Let G be a group and n be an integer. What does it mean to say that |g| = n for

g ∈ G? B) (5) Let I be an ideal in a ring R. What does it mean to say that a ≡ b mod I? C) (5) Give an example of an ideal in a ring R that is prime but not maximal. II. (10) Let R be the ring of functions f : R → R that have derivatives of all orders. Show that I = {f ∈ R : f(0) = f′(0) = 0} is an ideal in R. III. All parts of this problem refer to p(x) = x^{2} + x in Z_{2}[x]. A) (5) How many distinct cosets of the ideal (p(x)) are there in Z_{2}[x]? B) (10) Construct the coset addition and multiplication tables for the quotient ring

Z_{2}[x]/(p(x)).

C) (20) State and prove the First Isomorphism Theorem for rings. D) (10) Let Z_{2} × Z_{2} = {(a, b) : a, b ∈ Z_{2}}, a ring under the component-wise sum and

product operations. Show that f : Z_{2}[x] → Z_{2} × Z_{2} defined by f(g(x)) = (g(0), g(1)) is a ring homomorphism and determine ker(f).

E) (5) Deduce that Z_{2}[x]/(p(x)) is isomorphic as a ring to Z_{2} × Z_{2}. IV. (10) Let G = {4, 8, 12, 16} ⊂ Z_{20}. Is G a group under multiplication mod 20? If so, say why and determine if G is cyclic. If not, identify which of the group axioms fail. V. A) (10) State Lagrange’s Theorem for groups. B) (5) Let G be a group of order 60. Is it possible for G to contain elements a, b with

|a| = 3 and aba^{−1} = b^{4}? Why or why not?