Binary Operations - Introduction to Abstract Algebra - Exam, Exams for Algebra. Shree Ram Swarup College of Engineering & Management

Algebra

Description: This is the Exam of Introduction to Abstract Algebra and its key important points are: Binary Operations, Mapping, Common Factor, Highest Common Factor, Pair of Integers, Euclidean Algorithm, Different Pairs, Divisibility, Principal Residues, Multiplication Tables
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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 1 EXAMINATIONS, JANUARY/FEBRUARY 2010
MA20310 - Introduction to Abstract Algebra
Time allowed - 2 hours
Full marks will be given for complete answers to all questions in section A and to
three questions in section B. In section B, credit will be given to the best three
questions answered.
Calculators are not permitted.
18/12/2009
MA20310 - Introduction to Abstract Algebra 2 of 6
Section A
1. (a) Let Aand Bbe sets. Give the definitions of a mapping F:AB, and of a
binary operation on A. [3 marks]
(b) Let and be binary operations on Z×Nas follows:
(k, l)(m, n) = (kn +lm, ln)
(k, l)(m, n) = (km, ln)
i) Compute (7,2) (3,4) and (7,3) (4,2). [2 marks]
ii) Show that, for all a, b, c Z×N, (ab)c=a(bc).[4 marks]
iii) Find uZ×Nsuch that ua=a, for all aZ×N. [2 marks]
2. (a) Define a common factor of aand b, and the highest common factor HCF (a, b)
of a pair of integers a, b. [2 marks]
(b) Use the Euclidean Algorithm to determine HCF (165,66) and express this in
the form 165 ·x+ 66 ·yfor x, y Z. [5 marks]
(c) Find two different pairs (x, y)Z2of solutions to the equation
165 ·x+ 66 ·y= 99.
[3 marks]
3. Prove the following results about divisibility in Z:
(a) If b|athen b|(a).[2 marks]
(b) If b|aand c|bthen c|a. [2 marks]
4. Find the principal residues of
i) 120 mod 19,ii) 246123 mod 9,iii) 5431003 mod 5.
[8 marks]
5. (a) Write out the multiplication tables of Z7and Z8. [4 marks]
(b) Determine the units of Z7and Z8. [2 marks]
(c) Find the solution of 5x7
3 in Z7. [1 mark]
(d) Find the solution of a38
0. [1 mark]
(e) Are there solutions to the equation 2x8
3 in Z8? [1 mark]
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MA20310 - Introduction to Abstract Algebra 3 of 6
6. Find all integer solutions to the system of linear congruences:
3x7
4,5x8
2.
[8 marks]
7. Let p(x)F[x].
(a) Define the principal residue of [a(x)]p(x)for a(x)F[x]. [2 marks]
(b) Determine the principal residue of [x5]x2+x+2 in R[x]. [4 marks]
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MA20310 - Introduction to Abstract Algebra 4 of 6
Section B
8. State and prove the Fundamental Theorem of Arithmetic.
[20 marks]
9. (a) State Fermat’s Little Theorem and use it to compute the principal residue of
4970 mod 23. [6 marks]
(b) If pis prime, show that xpxhas a factorization into linear factors in Zp[x].
[8 marks]
(c) Using this factorization show that (p1)! p
≡ −1.[6 marks]
10. The quaternions Hmay be represented as the set of complex 2 ×2 matrices of the
form
ζ=t I +xi+yj+zk
=t+iz y +ix
y+ix t iz ,
with t, x, y, z R. The addition and multiplication operations on Hare taken to
correspond to the usual ones for their matrix representation.
(i) Show that i2=j2=k2=Iand
ijk =I.
[5 marks]
(ii) The quaternionic conjugate of ζis ¯
ζ=tI xiyjzk. Compute the
quaternionic modulus squared |ζ|2=ζ¯
ζand show that it is equal to the determinant
of the matrix representation of ζ.
[5 marks]
(iii) Show that every non-zero quaternion has an inverse.
[5 marks]
(iv) Noting the identity
exi+yj+zk= cos r I +sin r
r(xi+yj+zk),
where r=px2+y2+z2, show that
|exi+yj+zk|= 1.
[5 marks]
18/12/2009
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