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:A→B, 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, (a⊕b)⊕c=a⊕(b⊕c).[4 marks]

iii) Find u∈Z×Nsuch that u⊗a=a, for all a∈Z×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]

18/12/2009

##### Document information

Uploaded by:
aradhana

Views: 1181

Downloads :
0

Address:
Mathematics

University:
Shree Ram Swarup College of Engineering & Management

Subject:
Algebra

Upload date:
14/02/2013