2006 december

2006 december - Examiner: Professor E. Goren Associate...

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Unformatted text preview: Examiner: Professor E. Goren Associate Examiner: Professor H. Darmon 91:59” 6. McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH 235 ALGEBRA 1 Date: Friday December 22, 2006. Time: 9:00 AM — 12:00 PM INSTRUCTIONS . Please answer all question in the exam booklets provided. . This exam has two parts. The maximum grade you can obtain in this exam towards the calculation of your final course grade is 102 points. Follow the the instructions given in each section. Write you answers clearly. This is a closed book exam. Notes or books are not permitted. Calculators are not permitted. I Use of a regular and or translation dictionary are not permitted. This exam comprises the cover page, 1 page of Information regarding this exam and 3 pages of questions. MATH235 - Algebra I Final Exam — December 22, 2006 The Exam has two parts. The maximum grade you can obtain in this exam towards the calculation of your final course grade is 102 points. Follow the instructions given in each section. The use of calculators, notes or books is not allowed. Write your answers clearly. As a. rule, the notation is the same we used in class, or the class notes. In particular, Sn denotes the symmetric group of the set {1, 2, . . . ,n}; Z, Q, R and (C stand for the integers, rational, real and complex numbers, respectively. If n is a positive integer we denote the ring of congruences modulo n by Zn. 1le is a field then is the ring of polynomials with coefficients in IF and ]F[e] is the ring of dual numbers we have studied (62 = 0, etc); if R is a commutative ring M2(R) is the ring of two-by—two matrices with entries in R, under matrix addition and multiplication. Good luck! 2- MATH235 - ALGEBRA 1, FINAL EXAM — DECEMBER 22. 2006 Part A. {(36 points) In this part you need to give the correct answer with no explanations or proof. There is only one correct answer to each question. Each question is worth 4 points. A correct answer gives you 4 points; no answer gives you 0 points; wrong answer gives you —1 points. ( 1) The group of permutations of five elements S5 has an element of order 7. (a) Yes. (b) No. (2) The god of the polynomials f = m4 + 551: + 1 and g(x) = x2 — 1 in Z7[a:] is: (a) x2 — 1. (b) a: — 1. (c) 1. (d) a: + 1. (3) Any two groups with 6 elements are isomorphic. (a) Yes. (b) No. (4) The ring (QM/($3 — 1) is a field. Yes. (b) No. (5) Let n 2 1 be an integer. Every homomorphism of groups f : Z, —> Z is the zero homomorphism. (a) Yes. (b) No. (c) Depends on n. (6) 2144 is congruent to: (a) 1 (mod 13). (b) 2 (mod 5). (7) Let G be a group. The relation on G defined by “a N b if there exists 9 6 G such that gag*1 = b” is an equivalence relation. (a) Yes. (b) No. MATH235 - ALGEBRA I, FINAL EXAM — DECEMBER 22, 2006 3 (8) There exists a transitive action of S4 on the set {1, 2, 3, 4, 5}. (Recall that a tran- sitive action of a group G on a set S is an action such that the following holds: for every $1, 32 E S there exists 9 E G such that g * 31 = 32.) (a) Yes. (b) No. (9) Let G be a group and a, b E G be two elements of order 2, i.e., a2 = e and b2 = 6. Then (ab)2 = e. (a) Yes. (b) No. 4 MATH235 - ALGEBRA 1, FINAL EXAM — DECEMBER 22, 2006 Part B. (66 points) Answer the following questions. Give detailed proofs and state clearly which theorems, lemmas etc. you are using. (1) Let G be a finite group acting on a finite non—empty set S. The action of g E G on s E S is denoted g * 3. Define for g E G, [(9) = [{s E S : g * s = Prove the Cauchy—Frobenius formula: Let N be the number of orbits of G in S then N = TCITI ZgEGI(g)' (2) Find the number of roulettes with 12 sectors, 6 of which are red and 6 of which are black. (b) Find the number of necklaces with 12 stones, 6 of which are red and 6 of which are black. (3) Let m, n be relatively prime positive integers. Prove the Chinese Remainder Theorem: Zmn g Zm X Z; (isomorphism of rings). (b) State the analogous theorem for the ring of polynomials lF[:1:], where IF is a field. (4) Let f E be an irreducible polynomial with coefficients in IF and of degree 2 1. (a) Prove that L := is a field. i (b Prove that f has a root in L. ( ) (5) Prove that there are infinitely many prime numbers. ) b Prove that every positive integer is a product of prime numbers. (Note: No need to prove uniqueness. (ii) This doesn’t require the first part.) (6) State and prove the first isomorphism theorem for groups. (b) Prove that there is no surjective homomorphism 5'3 —> Z3. ...
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2006 december - Examiner: Professor E. Goren Associate...

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