Question

151 12. There are five multiple choice questions on this page. Three marks each. No partial marks. There is only one correct answer for each question. Circle the correct answer. (i) Consider the subgroup H = ([16]) of the additive group Z40- Which of the following left cosets of H is equal to [7] + H ? (A) (17) + H (B) (18) + H (C) [28] + H (D) [39] + H (E) [29] + H (ii) What is the inverse of [5] in the multiplicative group of units U43 ? (A) 39 (B) 37 (C) 25 (D) 19 (E) 26 (iii) Consider the addition and multiplication defined in Z, for the given value of n. Which of the following sets is not an integral domain? (A) {[0],[3], [6], 19), [12]} C Z15 (B) {[0), (4), (8), (12), [16]} cZ2 (C) {[0],[5],[10],[15] [20]} CZ (D) {[0],[6), (12), [18], [24]} c Z30 (E) {[0],[8], [16], [24], [32]} Zgo (iv) Suppose that pr) is an irreducible polynomial in Z w ith deg p(I) = 5, and z (a) is the field that is obtained by adjoining a zero a of p() to Z. How many elements does Z(a) have? (A) 52 (B) 72 (C) 35 (D) 57 (E) 70 (v) Consider the additive group Z. = {0,1,2,3), and the multiplicative groups of units in Z. Z., Zjo and Z12, which are Us = {1,2,3,4}, Ug = {1,3,5,7), U10 = {1,3,7,9}, and U12 = {1,5,7,11), respectively. Which of the following statements is FALSE? (et" means "isomorphic") (A) Us 2 U12 (B) UsU10 (C) Us 2 U (D) U19 2 (E) USZ

Answer 1

ů [ option of Ho <[16]> = 416, 32, 8,24, 03 [474H = {23, 39, 15, 31, 73 $o []+H= [39] ++ option o i) [5] • [26] = [130]s = [] [] so inverse of [5] is 26 option E jis option c [5] · [10] = 0 but both are non-zero. is correct iv) e option E v zu = 40, 1,2,33 - UE= {1,2,3,43 is cyclic Ug = {1,3,5,73 every element is of order o is cyclic , 42 ~ Ug do correct option is → (A) option A