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2004 december - MoGill UN iVERSlT Y FACULTY OF SCIENCE...

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Unformatted text preview: MoGill UN iVERSlT Y FACULTY OF SCIENCE FINAL EXAMINATION MATH 235a BASIC ALGEBRA l Examiner: Professor H. Darmon Date: Tuesday December 7, 2004 Associate Examiner: Professor A.Miasnikov Time: 2:00 P.M A 5:00 P.M lNSTRI ICTTONS This exam consists of the cover page and 2 pages of 12 questions All questions are to be answered in the Examination Booklet Calculators are allowed This is a closed book exam Notes are not permitted Definition Dictionaries are not allowed. 954‘?pr 189—235A: Basie Algebra 1 Final Exam Tuesday, December 7; 2004 This cream has twelve questions, iiior'tli. 9 points each. The final grim/(I will lfl taken out {#100, although it is possible to achieve a mania-mum grow o_/‘ NM. 1. Let (Uni-n30 be the Fibonacci sequence. defined recursively by the rule 1m : 0, U} = 1, "u.“ H I on + ll.n_1, for n. 7: 1. Show that an 5 2” for all n. 2 U. 2. Solve the congruence equation 6:1." 2*: 17 (mod 35). 3. Using only the basic properties of the god that were shown in (class. le'm'r' that the following; statement is true when p is prime: p divides (it) :> p divides a or p divides l). 4. Let Q denote the field of rational numbers. Show that the ggreaiesl common divisor of the polynomials 1:3 — 17 and $2 + 1 in Elli] is equal in l. and express 1 as a linear combination of these two polynomials. 5. Show that the ring Q[:r;]/ (:1:3 — 17) is afield. \-"V"'I'ite down the niiill iplir-atix‘e inverse of the element [1'2 + 1] in this field. 6. Give an example of two finite rings H1 and R2 which have the some cardinality but are not isornorphie. (You should justify your assertion.) 7. Show that the ring Rl:r.:]/ (If?) is not ismnorphic to the (flartesian prmluvt R X R of the real IlllIIll)el'S with itself. (EQ—L35A BaSlC Alafbm I Tuesdfifii fDé’q'onbfr 33 2001+ 8. Show that the ring R. = Zglxl/(I3 +1: + l) is a field with 8 ele111enis. 9. Give the ('leiinition of a. maximal ideal and a prime ideal in a. ('()11|1|111t;1ri\'e ring R. Show that it I is maximal, then it, is prime. Give an example to show that the converse does not hold. 10. Write down three. non—ismnorphie groups of order 8. (You do not need to justify your claim that these groups are non—isomorphic.) 11. Show that the group V 2 {id (l 2)(34). (l3)(24)._ (1—H) 23 )} is a normal subgroup of the symmetric group 64 of permutations on {i 2 3.3-1}. \\l1al group is the quotient group 5'4/‘1/ isomorphic to? (You int-13 ”ix-er the answer to this last question Without proof.) 12. Let F be a field. State (without proof) whether the tellmving a-isseriions are true or false. :1. The set 1 of constant polynomials in the ring F[.rl is an ideal oi" tip: 13. The set I of polynomials with constant term equal to zero is an ideal oi F[1l C. Every ideal in Z or in Flat] (Where F is a. field) is principal. (1. Every ideal in lel is prineip‘rrl. e. If G is a. finite group having an abelian normal subgroup 1' 1’ sin-I1 tlmi G / H is ahelian, the group G itself is ahelian. f. Every subgroup of a commutative group is nm'mal. g. if R is a ring, then the set Rx of units of R (i.e.. elements possessing e1 1r1ultiplicative inverse) is a group under multiplication. ...
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