Mathematics 351 – Abstract Algebra I Makeup Midterm Exam 2 – November 19, 2007

Directions: Place all work you want to have considered for credit in the blue exam book. You have two hours 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. A) (5) Let G be a group and H be a subgroup of G. What is |H|? B) (5) Let R be a ring, I an ideal in R and a, b ∈ R. What is true about a − b if the

cosets a + I and b + I are equal? C) (5) Are there any examples of ideals I in rings that are maximal but not prime? Why

or why not? II. (10) Let R be the ring of 2 × 2 matrices with real entries. Is

I = \{( \begin{matrix}0 & 0 \\ 0 & r \end{matrix} ) : r ∈ R\}

an ideal in R? Why or why not? III. (20) State and prove the First Isomorphism Theorem for rings. IV. All parts of this problem refer to p(x) = x^{2} + 1 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) (10) Let R = {(a, b) : a, b ∈ Z_{2}}, with operations ⊕ and ⊗ defined by (a, b) ⊕ (c, d) =

(a + c, b + d) and (a, b) ⊗ (c, d) = (ac, ad + bc). You may assume without proof that R is a commutative ring with identity under these operations. Show that the mapping f : Z_{2}[x] → R defined by f(g(x)) = (g(1), g′(1)) is a ring homomorphism, where g′ denotes the derivative, computed with coefficients mod 2.

D) (5) Deduce that Z_{2}[x]/(p(x)) is isomorphic as a ring to R from part C. V. (7) Let G = {6, 12, 18, 24} ⊂ Z_{30}. Is G a group under multiplication mod 30? If so, say why. If not, identify which of the group axioms fail. VI. A) (10) State Lagrange’s Theorem for groups. B) (8) Show that if G is a finite group of order n, then G has elements of order k for

every positive integer k dividing n if and only if G is cyclic.