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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 twoby—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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