Algebra and Number Theory Exercise Set Kamil Niedzia lomski 1 Algebra 1.1 Complex Numbers Exercise 1. Find real and imaginary part of complex numbers (1) 1 i 2+i (2) (3 + 7i)( 3 + i) (3) ( 3+i)( 1+i 3) (1+i) 2 (4) (2 + i)(3 i) + (2 + 3i)(3 + 4i) (5) (5+i)(4+i). 1+i Exercise 2. Find real numbers a, b such that (1) a(2 + 3i) + b(4 5i) = 6 2i a (2) + b+1 = 2 2 i 1+i (3) (2 + i)a + (1 + 2i)b = 1 4i (4) (3 + 2i)a + (1 + 3i)b = 4 9i. Exercise 3. Solve quadratic equations (1) z 2 = i (2) z 2 5z + 4 + 10i = 0 (3) z 2 (1 + i)z + 6 + 3i = 0 (4) z 2 + (2i 7)z + 13 i = 0. Exercise 4. Find the trigonometric form of complex numbers (1) i (2) 1 + i (3) 1 + i 3 (4) 3 i (5) 2 + 3 + i (6) cos α i sin α (7) sin α + i cos α (8) 1+itgα 1 tgα. Exercise 5. Compute (1) (1 + i) 1000 1
(2) 3 + i) 30 (3) ( 1 i 3 1+i ) 12 (4) (2 2 + i) 12. Exercise 6. Compute (1) 6 i (2) 4 8 3i 8 (3) 8 16 (4) 3 1 + i (5) 3 2 2i. Exercise 7. Describe the following sets on a complex plane (1) A 1 = {z C : z 1 i < 1} (2) A 2 = {z C : 2 < z < 3} (3) A 3 = {z C : 1 < Reiz < 0} (4) A 4 = {z C : Imz = 1} (5) A 5 = {z C : z 1 + z + 1 = 3} (6) A 6 = {z C : z 3 = z + i } (7) A 7 = {z C : re(z 2 ) = 2, (Im(z + i)) 2 = 1} 1.2 Groups, subgroups, Symmetric Group Exercise 8. Which of the following sets are groups (1) n th order complex roots of unity with multiplication. (2) { 1, 1} with multiplication. (3) Set of complex numbers with absolute value 2 with multiplication. (4) G = {7n : n N} with addition. (5) Set of numbers of the form a 2 + b 3, a, b Q, with addition. Exercise 9. Is (Z, ), where a b = a + b + 2, a group? Exercise 10. Consider all isometries of (1) an equilateral triangle, (2) a square, (3) circle. Which of above sets is a group with composition of functions? Exercise 11. Which of the following sets of matrices is a group (1) Set of symmetric matrices 2 2 with addition. (2) Set of nondegenerate matrices with addition. (3) Set of nondegenerate martices with multiplication. (4) Set of diagonal matrices with multiplication. (5) Set of matrices with determinant equal to 1 with multiplication. (6) Set of nonzero matrices of the form [ x y y x ], x, y R, 2
with multiplication. Exercise 12. Let G = [0, 2) and a b = a + b 2. Is (G, ) a group? Exercise 13. Prove that if every element a in a group G satisfies a 2 = e, where e is a neutral element, then G ia abelian. Exercise 14. Find all subgroups of the group of rotations of a square. Exercise 15. Find all subgroups of the group of isometries of a square. Exercise 16. In a set E = {(a, b) : a R \ {0}, b R} we define (a, b) (c, d) = (ac, ad + b). Show that (E, ) is a group. Show that F = {(a, b) E : b = 0} is a subgroup of E. Exercise 17. Compute στ, τσ, τ 1 σ 2, στσ, where ( ) ( 1 2 3 4 5 6 1 2 3 4 5 6 σ =, τ = 2 5 3 1 6 4 3 1 4 2 5 6 Find permutation ξ such that τξσ = ρ, where ( ) ( 1 2 3 4 1 2 3 4 σ =, τ = 2 4 3 1 4 2 1 3 ). ) ( 1 2 3 4, ρ = 2 1 4 3 Exercise 18. Find n such that σ n = e, where e denotes identity and σ = ( 1 5 ) ( 2 4 6 ) ( 3 7 8 9 ) 1.3 Homomorphisms of groups Exercise 19. Which of the following maps are homomorphisms of additive group Z? (1) φ(n) = n (2) φ(n) = 2n + 1 (3) φ(n) = n (4) φ(n) = n. Exercise 20. Which of the maps are homomorphisms of the group (R \ {0}, )? (1) φ(x) = 3x (2) φ(x) = x 2 (3) φ(x) = 1 x. Exercise 21. In the set G = (1, ) we define action a b = ab a b 2. Show that (A, ) is a group? Is this group isomorphic to (R +, )? Exercise 22. Are groups (R +, ) and (R, +) isomorphic? Exercise 23. Show that the group of all isometries of a square is isomorphic to symmetric group S 4. Exercise 24. Show that the group F from Exercise 16 is isomorphic to (R \ {0}, ). 3 ).
1.4 Rings Exercise 25. Which of the following sets with addition and multiplication is a ring? (1) A 1 = {a + b 2 : a, b Z}. (2) A 2 = {a + b 3 + c 5 : a, b, c Q}. (3) A 3 = {n Z : n = 2k or n = 3k for some k Z}. Exercise 26. Let (A, +, ) be a ring. In the set B = A Z we define (x, n) (y, m) = (x + y, n + m), (x, n) (y, m) = (xy + x + y, nm). Is (B,, ) a ring? Exercise 27. Find all zero divisors in the ring Z 36. Exercise 28. Which of the following sets of matrices (with addition and multiplication of matrices) is a ring? (1) Set of all symmetric matrices. (2) Set of all 3 3 uppertriangular matrices. (3) Set of all matrices of the form [ x y ay x where a Z is a fixed number. Exercise 29. Find zero divisors in the ring of 2 2 matrices. ], 2 Number Theory 2.1 Divisibility, Induction Exercise 30. Show that 120 divides n(n + 1)(n + 2)(n + 3)(n + 4) for any n N. Exercise 31. Prove by induction that for all n N (1) 5 (n 5 n) (2) 7 (n 7 n) (3) 9 (4 n + 15n 1) (4) 64 (3 2n+3 + 40n 27). Exercise 32. Prove that n 2 divides (n + 1) n 1 for every n N. Exercise 33. Show that the sum of 2n + 1 consecutive integers is divisible by 2n + 1. Exercise 34. Prove that 6 digit number of the form abcabc, a, b, c are the digits, is divisible by 7, 11 and 13. Exercise 35. Prove that for all n N, 133 (11 n+2 + 12 2n+1 ). Exercise 36. Prove that for any n N, n 2 + 2 is not divisible by 4. 4
2.2 Gratest Common Divisor, division with reminder, Euclid Algorythm Exercise 37. Using Euclid Algorythm compute gratest common divisor of the following numbers (1) 963 and 657 (2) 423 and 198 (3) 2947 and 3997 (4) 2689 and 4001. Exercise 38. When n is divided by 9 the reminder is 5. What is the reminder of the division of n(n 2 + 7n 2) by 9? Exercise 39. Prove that for all n N, 15 n divided by 7 gives reminder 1. Exercise 40. Find gratest common divisor of 1819 and 3587 and find x, y Z such that 1819x + 3587y = 17. Exercise 41. Prove that there aren t any interers x, y such that x + y = 100 and gcd(x, y) = 3. Exercise 42. Find all pairs of neutral numbers (x, y) satisfying x+y = 100, gcd(x, y) = 5. Exercise 43. Find integer solutions to the following equations (1) 5x + 4y = 21, (2) 17x + 13y = 181, (3) 5x 2y = 1, (4) 6x + 7y = 59, (5) 10x + 7y = 97, (6) 4x + 9y = 91, (7) 19x + 23y = 3, (8) 47x 25y = 279, (9) 963x + 657y = 243. Exercise 44. Find x, y N such that (1) 24x + 15y = 9, (2) 126x 102y = 18, (3) 13x + 25y = 265. 2.3 Congruences Exercise 45. Find last digit of the number 2 1000. Exercise 46. Show that 61! 63! mod 42 Exercise 47. Show that if n is odd then n 2 1 0 mod 8. Exercise 48. Solve x 100 1 mod 7. 5
Exercise 49. Show that if p is a prime number and a 2 b 2 or p divides a b (a, b Z). mod p, then p divides a + b Exercise 50. Show that (1) 3 80 + 7 80 2 mod 5, (2) 3 80 + 7 80 2 mod 100. Exercise 51. Solve the following congruences (1) 5x 2 15x + 22 0 mod 3, (2) 3x 1 mod 5, (3) 8x 3 mod 14, (4) x 2 + x + 1 0 mod 2, (5) 6x 3 mod 9, (6) 5x 3 mod 12. 2.4 Prime Numbers Exercise 52. Show that if p is a prime number grater than 5, then p 2 divided by 30 gives the reminder 1 or 19. Exercise 53. Show that for any n N, n > 1, n 4 + 4 is not prime. Exercise 54. Find smallest number of the form 3 n + 2 which is not prime. Exercise 55. Find neutral number n such that f(n) is not a prime number, where (1) f(n) = n 2 + n + 17, (2) f(n) = n 2 + 21n + 1, (3) f(n) = 3n 2 + 3n + 23. 6