Modern Algebra. Previous year Questions from 2017 to Ramanasri

Transcription

1 Moder Algebra Previous year Questios from 017 to 199 Ramaasri 017 S H O P NO- 4, 1 S T F L O O R, N E A R R A P I D F L O U R M I L L S, O L D R A J E N D E R N A G A R, N E W D E L H I. W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : /6363/6464 1

2 Let G be a group of order. Show that G is isomorphic to a subgroup of the permutatio group S. [10 Marks]. Let F be a field ad [ ] f X, g X F X Fxdeote the rig of polyomial over F i a sigle variable X. For with gx ( ) 0, show that there exist q( X ), r( X ) F[ X ] such that degree rx ( ) degree gx ( ) ad f ( X) q( X). g( X) r( X). [0 Marks] 3. Show that the groups Z 5 Z 7 ad Z 35 are isomorphic. [15 Marks] Let K be a field ad KX [ ] be the rig of polyomials over K i a sigle variable X for a polyomial f K[ X] Let (f) deote the ideal i KX [ ] geerated by f. show that (f) is a maximal ideal i KX [ ] if ad oly iff is a irreducible polyomial over K. (10 marks). Let p be prime umber ad Z p deote the additive group of itegers modulo p. show that that every o- zero elemet Z of geerates Z p p (15 marks) 3. Let K be a extesio of a field F prove that the elemet of K which are algebraic over F form a subfield of K Further if F K L Fare fields L is algebraic over K ad K is algebraic over F the prove that L is algebraic over F. (0 marks) 4. Show that every algebraically closed field is ifiite. (15 marks) (i) How may geerators are there of the cyclic group G of order 8? Explai. (5 Marks) e, a, b, c of order 4, where e is the idetity, costruct compositio tables (ii) Takig a group showig that oe is cyclic while the other is ot (5 Marks) 6. Give a example of a rig havig idetity but a subrig of this havig a differet idetity. 7. If R is a rig with uit elemet 1 ad is a homomorphism of R oto R', prove that (1) is the uit elemet of R ' 8. Do the followig sets form itegral domais with respect to ordiary additio ad multiplicatio? Is so, state if they are fields: (5+6+4=15 Marks) (i) The set of umbers of the form b withb ratioal. (ii) The set of eve itegers. (iii) The set of positive itegers. 014 x y 9. Let G be the set of all real where xz 0 matrices where. Show that G is group uder 0 z 1 matrix multiplicatio. Let N deote the subset a : a R. Is N a ormal subgroup ofg? 0 1 Justify your aswer. Reputed Istitute for IAS, IFoS Exams Page

3 10. Show that Z7 is a field. The fid [5] [6] 1 ad [4] 1 i Z Show that the setab: 1, where a adb are real umbers, is a field with respect to usual additio ad multiplicatio. 1. Prove that the set Q( 5) a b 5 : a, b Q is commutative rig with idetity. 013 a b 13. Show that the set of matrices S a, br is a field uder the usual biary operatios of b a matrix additio ad matrix multiplicatio. What are the additive ad multiplicative idetities ad what is the iverse of 1 1 a b? Cosider the map f : C S defied by f( a ib). Show 1 1 b a that f is a isomorphism. (Here R is the set of real umbers ad C is the set of complex umbers) 14. Give a example of a ifiite group i which every elemet has fiite order 15. What are the orders of the followig permutatio i S 10? ad What is the maximal possible order of a elemet i S 10? Why? Give a example of such a elemet. How may elemets will there be i S 10 of that order? (13 Marks) 17. Let J a ib / a, b Z be the rig of Gaussia itegers (subrig ofc ). Which of the followig is J : Euclidea domai, pricipal ideal domai, ad uique factorizatio domai? Justify your aswer C 18. Let R rig of all real value cotiuous fuctios o[0, 1], uder the operatios C 1 ( f g) x f( x) g( x), ( fg) x f( x) g( x). Let M f R / f 0. Is M a maximal ideal of R? Justify your aswer How may elemets of order are there i the group of order 16 geerated by a ad b such that 1 the order of a is 8, the order ofb is adbab a 1. (1 Marks) 0. How may cojugacy classes does the permutatio group S 5 of permutatio 5 umbers have? Write dow oe elemet i each class (preferably i terms of cycles). 1. Is the ideal geerated by ad X i the polyomial rig ZX [ ] of polyomials i a sigle variable X with coefficiets i the rig of itegers Z, a pricipal ideal? Justify your aswer Z [ i] a ib / a, b Z.. Describe the maximal ideals i the rig of Gaussia itegers Reputed Istitute for IAS, IFoS Exams Page 3

4 3. Show that the setg f, f, f, f, f, f of six trasformatios o the set of Complex umbers z 1 1 ( z1) defied by f 1 ( z) z, f ( z) 1 z, f 3 ( z), f 4 ( z), f 5 ( z), f 6 ( z) (1 z) z (1 z) z is a oabelia group of order 6 w.r.t. compositio of mappigs (1 Marks) 4. Prove that a group of Prime order is abelia. (6 Marks) 5. How may geerators are there of the cyclic group ( G,.) of order 8? (6 Marks) 6. Give a example of a group G i which every proper subgroup is cyclic but the group itself is ot cyclic 7. Let F be the set of all real valued cotiuous fuctios defied o the closed iterval[0, 1]. Prove that( F,,.) is a Commutative Rig with uity with respect to additio ad multiplicatio of fuctios defied poit wise as below: f g x f( x) g( x) x [0, 1] where f, g F ad fg x f( x) g( x) 8. Let a ad b be elemets of a group, with a e, b 6 4 e ad ab b a. Fid the order of ab, ad m express its iverse i each of the forms a b adb m a 9. Let G R be the set of all real umbers omittig -1. Defie the biary relatio * o G by a b a b ab. Show( G, ) is a group ad it is abelia (1 Marks) 30. Show that a cyclic group of order 6 is isomorphic to the product of a cyclic group of order ad a cyclic group of order 3. Ca you geeralize this? Justify. (1 Marks) 31. Let( R,.) be the multiplicative group of o-zero reals ad( GL(, R), X ) be the multiplicative GL(, R) group of o-sigular real matrices. Show that the quotiet group SL(, R) ad( R,.) are SL(, R) A GL(, R) / det A 1 what is the ceter of GL(, R ) isomorphic where 3. LetC f : I [0, 1] R / f is cotiuous. Show C is a commutative rig with 1 uder poit wise additio ad multiplicatio. Determie whetherc is a itegral domai. Explai Cosider the polyomial rig Qx [ ]. Show p( x) x is irreducible overq. Let I be the ideal Qx [ ] Qx [ ] i geerated by px ( ). The show that is a field ad that each elemet of it is of the form I a0 a1t at witha 0, a 1, a i Q adt x I 34. Zi [] Show that the quotiet rig 1 3i is isomorphic to the rig where Zideotes [] the rig of 10 ZZ Gaussia itegers If R is the set of real umbers ad R is the set of positive real umbers, show that R uder additio ( R, ) ad R uder multiplicatio (,.) are isomorphic. Similarly ifq is set of ratioal R umbers ad Q is the set of positive ratioal umbers, are ( Q, ) ad( Q,.) isomorphic? Justify your aswer. (4+8=1 Marks) Reputed Istitute for IAS, IFoS Exams Page 4

5 36. Determie the umber of homomorphisms from the additive group Z 15 to the additive group Z 10 ( Z is the cyclic group of order ) (1 Marks) 37. How may proper, o-zero ideals does the rig Z 1 have? Justify your aswer. How may ideals does the rig Z1 Z1 have? Why? (+3+4+6=15Marks) 38. Show that the alteratig group of four letters A4 has o subgroup of order Show that ZX [ ] is a uique factorizatio domai that is ot a pricipal ideal domai ( Z is the rig of itegers). Is it possible to give a example of pricipal ideal domai that is ot a uique factorizatio domai? ( ZX [ ] is the rig of polyomials i the variable X with iteger.) Z [ X] 40. How may elemets does the quotiet rig 5 have? Is it a itegral domai? Justify yours X 1 aswers Let R0 be the set of all real umbers except zero. Defie a biary operatio o R 0 as a b a b where a deotes absolute value of a. Does( R0, ) form a group? Examie. (1 Marks) 4. Suppose that there is a positive eve iteger such that a a for all the elemets a of some rig R. Show that aa0 for all a Rad a b 0 a b for all a, b R (1 Marks) 43. Let G adg be two groups ad let : G Gbe a homomorphism. For ay elemet a G (i) Prove that O ( a) O( a) (ii) Ker is ormal subgroup ofg. 44. Let R be a rig with uity. If the product of ay two o-zero elemets is o-zero. The prove that ab 1 ba 1. Whether Z6 has the above property or ot explai. Is Z6 a itegral domai? 45. Prove that every Itegral Domai ca be embedded i a field. 46. Show that ay maximal ideal i the commutative rig Fxof [ ] polyomial over a field F is the pricipal ideal geerated by a irreducible polyomial If i a group G, a 5 e, e is the idetity elemet ofg aba 1 for a, bg, b the fid the order of b (1 Marks) 48. a b Let R c d where a, b, c, d Z. Show that R is a rig uder matrix additio ad multiplicatio a 0 A, a, bz. The show that A is a left ideal of R but ot a right ideal of R. b 0 (1 Marks) 49. (i) Prove that there exists o simple group of order 48. (ii) 1 3 ad Z 3is a irreducible elemet, but ot prime. Justify your aswer. 50. Show that i the rig R a b 5 a, b Z prime, but y ad y have o g.c.d i,. The elemet 3 ad 1 5 are relatively R where 71 5 (30 Marks) Reputed Istitute for IAS, IFoS Exams Page 5

6 Let S be the set of all real umbers except -1. Defie o S by a b a b ab. Is ( S, ) a group? Fid the solutio of the equatio x 3 7 i S. (1 Marks) 5. IfG is a group of real umbers uder additio ad N is the subgroup ofg cosistig of itegers, prove that G is isomorphic to the group H of all complex umbers of absolute value 1 uder N multiplicatio (1 Marks) 53. (i) Let OG ( ) 108. Show that there exists a ormal subgroup or order 7 or 9. (ii) LetG be the set of all those ordered pairs( a, b) of real umbers for which a 0 ad defie i G, a operatio as follows:( a, b) ( c, d) ( ac, bc d) Examie whetherg is a group w.r.t the operatio. If it is a group, is G abelia? 54. Show that Z a b : a, bz is a Euclidea domai. (30 Marks) If M ad N are ormal subgroups of a groupg such that M N e, show that every elemet of M commutes with every elemet of N. (1 Marks) 56. Show that(1 i) is a prime elemet i the rig R of Gaussia itegers. (1 Marks) 57. Let H ad K be two subgroups of a fiite groupg such that H G ad K G H K e.. Prove that 58. ' If f : G G is a isomorphism, prove that the order agof is equal to the order of fa ( ) 59. Prove that ay polyomial rig Fx [ ] over a field F is U.F.D (30 Marks) If p is prime umber of the form 4 1, beig a atural umber, the show that cogruece x 1(mod p) is solvable. 61. LetG be a group such that of all a, b G (i) ab ba (ii) O( a), O( b) 1 the show that (1 Marks) O( ab) O( a) O( b) (1 Marks) 4 6. Verify that the set E of the four roots of x 10forms a multiplicative group. Also prove that a trasformatio T, T( ) i is a homomorphism from I (Group of all itegers with additio) oto E uder multiplicatio. 63. Prove that if cacellatio law holds for a rig R the a( 0) Ris ot a zero divisor ad coversely 64. Z The residue class rig is a field iff m is a prime iteger. ( m ) 65. Defie irreducible elemet ad prime elemet i a itegral domai D with uits. Prove that every prime elemet i D is irreducible ad coverse of this is ot (i geeral) true. (5 Marks) Reputed Istitute for IAS, IFoS Exams Page 6

7 66. If H is a subgroup of a group G such that subgroup ofg Show that the rig Z i a ib a, bz, i 1 x H for every xg, the prove that H is a ormal (1 Marks) of Gaussia itegers is a Euclidea domai 68. Let R be the rig of all real-valued cotiuous fuctios o the closed iterval[0, 1]. (1 Marks) 1 Let M f( x) R f 0 3. Show that M is a maximal ideal of R 69. Let M ad N be two ideals of a rig R. Show that M Nis a ideal of R if ad oly if either M N or N M 70. Show that Q( 3, i) is a splittig field for x 5 3x 3 x 3where Q is the field of ratioal umbers 71. Prove that x x 4 is irreducible over F the field of itegers modulo 11 ad prove further that Fx [ ] is a field havig 11 elemets. ( x x 4) 7. Let R be a uique factorizatio domai (U.F.D), the prove that Rx [ ] is also U.F.D Show that a group of order 35 is cyclic. (1 Marks) Show that polyomial 5x 9x 3x 3is irreducible over the field of ratioal umbers(1 Marks) 75. Show that a group of p is abelia, where p is a prime umber. 76. Prove that a group of order 4 has a ormal subgroup of order Prove that i the rig Fx [ ] of polyomial over a field F, the ideal1 px ( ) is maximal if ad oly if the polyomial px ( ) is irreducible over F. 78. Show that every fiite itegral domai is a field 79. Let F be a field with q elemets. Let E be a fiite extesio of degree over F. Show that E has q elemets Let K be a field ad G be a fiite subgroup of the multiplicative group of o-zero elemets of K. Show that G is a cyclic group. (1 Marks) 3 p Prove that the polyomial1 x x x... x where p is prime umber is irreducible over the field of ratioal umbers. (1 Marks) 8. Let N be a ormal subgroup of a groupg. Show that G is abelia if ad oly if for all N 1 x, yg, xyz N 83. If R is a commutative rig with uit elemet ad M is a ideal of R, the show that maximal ideal of R if ad oly if R is a field M 84. Prove that every fiite extesio of a field is a algebraic extesio. Give a example to show that the coverse is ot true. Reputed Istitute for IAS, IFoS Exams Page 7

8 85. Let be a fixed positive iteger ad let 000 Z be the rig of itegers modulo. LetG a Z a a 0 ad a is relatively prime to. Show that G is a group uder multiplicatio defied i Z. Hece, or otherwise, ( ) show that a a(mod ) for all itegers a relatively prime to where ( ) deotes the umber of positive itegers that are less tha ad are relatively prime to 86. Let M be a subgroup ad N a ormal subgroup of group G. Show that MN is a subgroup of G ad MN N is M isomorphic to M N. 87. Let F be a fiite field. Show that the characteristic of F must be a prime iteger p ad the umber of m elemets i F must be p for some positive itegerm. 88. Let F be a field ad Fxdeote [ ] the set of all polyomials defied over F. If fx ( ) is a irreducible Fx [ ] polyomial i Fx [ ], show that the ideal geerated by fx ( ) i Fxis [ ] maximal ad is a field. f( x) 89. Show that ay fiite commutative rig with o zero divisors must be a field If is a homomorphism of G itog with kerel K, the show that K is a ormal subgroup of G. 91. If p is prime umber ad p / O( G), the prove thatg has a subgroup of order p. 9. Let R be a commutative rig with uit elemet whose oly ideals are (0) ad R itself. Show that R is a field Prove that if a group has oly four elemets the it must be abelia. 94. If H ad K are subgroups of a groupg the show that HK is a subgroup of G if ad if oly HK KH. 95. Let( R,,.) be a system satisfyig all the axioms for a rig with uity with the possible exceptio of a b b a. Prove that ( R,,.) is a rig. 96. If p is prime the prove that Z p is a field. Discuss the case whe p is ot a prime umber. 97. Let D be a pricipal domai. Show that every elemet that its either zero or a uit i D is a product of irreducible Show that a ecessary ad sufficiet coditio for a subset H of a group G to be a subgroup is HH H. 99. Show that the order of each subgroup of a fiite group is a divisor of the order of the group. Reputed Istitute for IAS, IFoS Exams Page 8

9 100. I a group G, the commutator ( a, b) a, b G is the elemet 1 1 aba b ad the smallest subgroup cotaiig all commutators is called the commutator subgroup ofg. Show that a quotiet group G H is abelia if ad oly if H cotais the commutator subgroup ofg If x xfor all x i a rig R, show that R is commutative. Give a example to show that the coverse is ot true. 10. Show that a ideal S of the rig of itegers Z is maximal ideal if ad oly if S is geerated by a prime iteger Show that i a itegral domai every prime elemet is irreducible. Give a example to show that the coverse is ot true Let R be the set of real umbers adg ( a, b) a, b R, a 0. G G G is defied by ( a, b) ( c, d) ( ac, bc d). Show that( G, ) is a group. Is it abelia? 105. Let f be a homomorphism of a group G oto a group G ' with kerel H. For each subgroup K ' of G ' defie K by. Prove that (i) (ii) K ' is isomorphic to K H G ' is isomorphic to K G K ' 106. Prove that a ormal subgroup H of a group G is maximal, if ad oly if the quotiet group G H is simple I a rig R, prove that cacellatio laws hold. If ad oly if R has o zero divisors If S is a ideal of rig R ad T ay subrig of R, the prove that S is a ideal of S T s t ss, t T Prove that the polyomial x x 4 is irreducible over the field of itegers modulo Let G be a fiite set closed uder a associative biary operatio such that ab ac b c ad ba ca b c for all a, b, c G. Prove that G is a group LetG be group of order p, where p is a prime umber ad 0. Let H be a proper subgroup of G ad N( H) x G : x 1 hx H h H. Prove that N( H) H. 11. Show that a group of order 11 is ot simple Let R be a rig with idetity. Suppose there is a elemet a of R which has more tha oe right iverse. Prove that a has ifiitely may right iverses Let F be a field ad let px ( ) be a irreducible polyomial over F. Let px ( ) be the ideal geerated by px ( ). Prove that px ( ) is a maximal ideal Let F be a field of characteristic p 0. Let Fx ( ) be the polyomial rig. Suppose f x a a x a x a x ( ) is a elemet of ( ) 1 Fx. Defie f( x) a a x 3 a x... a x. If 1 3 p fx ( ) 0, prove that there exists g( x) F( x) such that f( x) g( x ). Reputed Istitute for IAS, IFoS Exams Page 9

10 IfG is a group such that( ab) a b for three cosecutive itegers for all a, b G, the prove that G is abelia Ca a group of order 4 be simple? Justify your claim 118. Show that the additive group of itegers modulo 4. Is isomorphic to the multiplicative group of the o-zero elemets of itegers modulo 5. State the two isomorphisms 119. Fid all the uits of the itegral domai of Gaussia itegers. 10. Prove or disprove the statemet: The polyomial rig Ixover [ ] the rig of itegers is a pricipal ideal rig. 11. If R is a itegral domai (ot ecessarily a uique factorizatio domai) ad F is its field of f0 quotiets, the show that ay elemet fx ( ) i Fx ( ) is of the form ( ) ( x fx ) where a f ( If G is a cyclic group of order ad p divides, the prove that there is a homomorphism of G oto a cyclic group of order p. What is the Kerel of homomorphism? 13. Show that a group of order 56 caot be simple. 14. Suppose that H, K are ormal subgroups of a fiite groupg with H a ormal subgroup of K. If K G P, S H H are isomorphic. Z[ 3] a 3 b : a, b Z is ot a uique factorizatio 15. If Z is the set of itegers the show that domai 16. Costruct the additio ad multiplicatio table for Z3[ x] x 1 ad x 1 is the ideal geerated by( x 1) i Z 3 [ x ] where Z 3 is the set of itegers modulo 3 1/ 1/3 17. Let Q be the set of ratioal umber ad Q(, ) the smallest extesio field of Q cotaiig 1/ 1/3 1/ 1/3,. Fid the basis for Q(, ) overq If H is a cyclic ormal subgroup of a group G, the show that every subgroup of H is ormal ig. 19. Show that o group of order 30 is simple If p is the smallest prime factor of the order of a fiite group G, prove that ay subgroup of idex p is ormal If R is uique factorizatio domai, the prove that ay f R[ x] is a irreducible elemet of Rx [ ], if ad oly if either f is a irreducible elemet of R or f is a irreducible polyomial i Rx [ ]. Reputed Istitute for IAS, IFoS Exams Page 10

11 13. Prove that x 1 ad x x 4 are irreducible over F, the field of itegers modulo 11. Prove also Fx [ ] Fx [ ] that ad are isomorphic fields each havig 11 elemets. x 1 x x Fid the degree of splittig field x 3x x 3over Q, the field of ratioal umbers. Reputed Istitute for IAS, IFoS Exams Page 11