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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 maniamum grow o_/‘ NM. 1. Let (Unin30 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 ipliratix‘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. ('()111111t;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 3.31}. \\l1al
group is the quotient group 5'4/‘1/ isomorphic to? (You int13 ”ixer the answer
to this last question Without proof.) 12. Let F be a ﬁeld. State (without proof) whether the tellmving aisseriions
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’ sinI1 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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 Spring '09
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 Algebra

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