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LANCASTER UNIVERSITY
2008 EXAMINATIONS
PART II (Second Year) MATHEMATICS & STATISTICS 2 hours Math 225: Groups and Rings
You should answer ALL Section A questions and THREE Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. SECTION A
A1. Let S = Q \ { 1 }. Show that a โ b = ab โ a โ b + 2 defines a binary operation on S, and that under this binary operation S is an abelian group. [8]
A2. The following is an incomplete Cayley table for a group G with five elements v, w, x, y and z. โ v w x y z v y w x y x z Explain which element must be the identity of G. Complete the Cayley table (you need not justify each step). [6]
please turn over
SECTION A continued
A3. Let G and H be groups, and ฯ : G โ H a function. (a) Define what it means for ฯ to be a group homomorphism. [1] (b) If ฯ is a group homomorphism and g โ G, show that (i) ฯ(eG) = eH , where eG and eH are the identity elements of G and H respectively; [2] (ii) ฯ(gโ^1 ) = ฯ(g)โ^1 for all g โ G. [2]
A4. Let ฯ = (^ 1 2 3 4 5 6 7 8 96 8 1 9 4 7 3 2 5^ )^ and ฯ = (2 4 8 5)(1 9 6 3)(2 9 3 4 8)(1 7 5 6).
(a) Write ฯ and ฯ in cycle notation. [2] (b) Write ฯ and ฯ as products of transpositions. [2] (c) Calculate sign(ฯ) and sign(ฯ). [2]
A5. (a) Define what it means for a ring R to be an integral domain. [2] (b) Let a, b and c be elements of an integral domain R. Suppose that a 6 = 0 and ab = ac. Show that b = c. [2] (c) Explain why the ring F ({ 1 , 2 }, R) of real-valued functions on the set { 1 , 2 } is not an integral domain. [3] (d) Decide whether or not the mapping ฯ : F ({ 1 , 2 }, R) โ R given by ฯ(f ) = f (1) + f (2) for each function f :{ 1 , 2 } โ R is a ring homomorphism; justify your answer. [3]
A6. (a) Find the characteristic of each of the rings^ Z 3 and^ Z 6.^ [2] (b) Explain why Z 3 and Z 6 cannot be isomorphic rings. [2]
A7. Find the multiplicative inverse of the element 19 inฬ Z 29 , and use this to solve the equation 19 ฬ x = ฬ7 in Z 29. [6]
A8. (a) Let^ M^ be an ideal in a ring^ R. What is meant by saying that^ M^ is a^ maximal^ ideal in R? [2] (b) Explain why the ideal 2Z of even integers is a maximal ideal in Z. (You do not need to prove that 2Z is an ideal in Z.) [3] please turn over
SECTION B continued
B3. Let UT 2 (R) =
a b 0 c
: a, b, c โ R
(a) Show that UT 2 (R) is a subring of M 2 (R). [3] (b) Define ฯ : UT 2 (R) โ M 2 (R) by
ฯ
( (^) a b 0 c
( (^) a 0 0 c
(a, b, c โ R). Show that ฯ is a ring homomorphism, and find its kernel and image. [6] (c) State the Fundamental Isomorphism Theorem for rings. [2] Define I =
0 a 0 0
: a โ R
(d) Show that I is an ideal in UT 2 (R), and identify explicitly a subring of M 2 (R) that is isomorphic to UT 2 (R)/I. Your answers to (a), (b), and (c) may be useful. [5] (e) Is the ring UT 2 (R)/I commutative? Justify your answer. [4]
please turn over
SECTION B continued
B4. Let Q[i] = {q + ri : q, r โ Q}. (a) Show that Q[i] is a subfield of C. [5] (b) Using the division algorithm or otherwise, find the quotient and remainder when dividing the polynomial f (X) = 2X^3 + 3X^2 โ X + 1 by X^2 + 1 in Q[X]. [2] Define I = {(X^2 + 1)p(X) : p(X) โ Q[X]}. You may assume without proof that I is an ideal in Q[X]. (c) Let f (X) = 2X^3 + 3X^2 โ X + 1 as in (b). Find a polynomial g(X) โ Q[X] of degree at most one such that fฬ (X) = gฬ(X) in Q[X]/I, where fฬ (X) and gฬ(X) denote the cosets in Q[X]/I of the polynomials f (X) and g(X) (as usual). (Hint: the result found in (b) may be useful.) [2] Define a mapping ฯ : Q[i] โ Q[X]/I by ฯ(q + ri) = qฬ + rX (q, r โ Q). (d) Show that ฯ is a ring homomorphism. [5] (e) Show that ฯ is injective. [2] (f) Show that ฯ is surjective. [3] (Hint: use the division algorithm and an argument similar to (c).) (g) Using (d)โ(f), explain why the ring Q[i] is isomorphic to Q[X]/I. [1]
end of exam
