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

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]

18/12/2009

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 xp−xhas a factorization into linear factors in Zp[x].

[8 marks]

(c) Using this factorization show that (p−1)! 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 −xi−yj−zk. 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