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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.

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MA20310 - Introduction to Abstract Algebra 2 of 6

Section A

1. (a) Let A and B be 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× N as 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× N such that u⊗ a = a, for all a ∈ Z× N. [2 marks]

2. (a) Define a common factor of a and 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 · y for x, y ∈ Z. [5 marks]

(c) Find two different pairs (x, y) ∈ Z2 of solutions to the equation

165 · x+ 66 · y = 99.

[3 marks]

3. Prove the following results about divisibility in Z : (a) If b|a then b| (−a) . [2 marks]

(b) If b|a and c|b then 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 Z7 and Z8. [4 marks]

(b) Determine the units of Z7 and Z8. [2 marks]

(c) Find the solution of 5x 7≡ 3 in Z7. [1 mark]

(d) Find the solution of a3 8≡ 0. [1 mark]

(e) Are there solutions to the equation 2x 8≡ 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:

3x 7≡ 4, 5x 8≡ 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 p is prime, show that xp − x has 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 H may 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 H are taken to correspond to the usual ones for their matrix representation.

(i) Show that i2 = j2 = k2 = −I and

ijk = I.

[5 marks]

(ii) The quaternionic conjugate of ζ is ζ̄ = tI − xi − yj − zk. Compute the quaternionic modulus squared |ζ|2 = ζζ̄ and 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 = √ x2 + y2 + z2, show that

|exi+yj+zk| = 1.

[5 marks]

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MA20310 - Introduction to Abstract Algebra 5 of 6

11. Let S be a ring with addition operation + and multiplication ·. Using only the ring axioms, prove that:

(a) Whenever a, b, c ∈ S satibfy a+ b = a+ c, then b = c. [2 marks]

(b) The identity element is unique. [2 marks]

(c) Let a ∈ S be a unit, then (a−1)−1 = a. [3 marks]

Let (T,+, ·) be a second ring. (d) Define what it means for a mapping ϕ : S 7→ T to be a ring homomorphism.

[3 marks]

(e) Show that ϕ maps the identity of S to the identity of T. [3 marks]

(f) Define what is meant by the term ideal of a ring. [3 marks]

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MA20310 - Introduction to Abstract Algebra 6 of 6

Axioms of Addition on a set F

A1: (closure) for all a, b ∈ F, a+ b ∈ F,

A2: (associativity) for all a, b, c ∈ F, (a+ b) + c = a+ (b+ c),

A3: (the zero) There exists a special element denoted by θ such that θ+ a = a+ θ = a, for all a ∈ F,

A4: For every a ∈ F there exists an element denoted as (−a) ∈ F such that a+ (−a) = θ = (−a) + a,

A5: (commutativity) for all a, b ∈ F, a+ b = b+ a.

Axioms of Multiplication on a set F

M1: (closure) for all a, b ∈ F, ab ∈ F,

M2: (associativity) for all a, b, c ∈ F, (ab) c = a (bc),

M3: (the identity) There exists a special element denoted by u such that u.a = a.u = a, for all a ∈ F,

M4: For every θ 6= a ∈ F there exists an element denoted as a−1 ∈ F such that a (a−1) = u = (a−1) a,

M5: (commutativity) for all a, b ∈ F, ab = ba.

Axioms of Distributivity

D1: (left) for all a, b, c ∈ F, then a(b+ c) = ab+ ac,

D2: (right) for all a, b, c ∈ F, then (a+ b)c = ac+ bc.

Definition: A set G is a group if there exists a binary relation (group law, or group product) ? on G such that

G1: g ? h ∈ G, for all g, h ∈ G,

G2: (g1 ? g2) ? g3 = g1 ? (g2 ? g3), for all g1, g2, g3 ∈ G,

G3: There exists an element e such that g ? e = e ? g = g for all g ∈ G,

G4: For each g ∈ G, there exists an element g−1 ∈ G such that g ? g−1 = e = g−1 ? g.

Definition: A set F is a field if there exists a pair of binary operations (addition and multiplication) satisfying A1−A5, M1−M5, (θ 6= u) respectively, and jointly satisfying D1−D2.

Definition: A ring (with unit) is a triple (S,+, ·) consisting of a set S, a binary operation + satisfying A1 − A5 on S, a binary operation · satisfying M1 −M3 on S, and both binary operations satisfying D1 − D2. If the multiplication · also satisfies M5 the we have a commutative ring.

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