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Flashcards

Abstract Algebra Review

Alyssa Baxter Mathematics English 85 cards

Abstract Algebra Review

Cards in this deck (85)

  1. 1Group definitionGroup definition
  2. 2abelianabelian
  3. 3Identity and Inverse Chapter 2 TheoremIdentity and Inverse Chapter 2 Theorem
  4. 4Cancellation and souls-shoes (ch 2)Cancellation and souls-shoes (ch 2)
  5. 5Subgroup DefinitionSubgroup Definition
  6. 6One-step subgroup testOne-step subgroup test
  7. 7Two-step subgroup testTwo-step subgroup test
  8. 8(the subgroup generated by a) is a _______ of Gsubgroup
  9. 9Center of a group def & theoremCenter of a group def & theorem
  10. 10Centralizer def & theoremCentralizer def & theorem
  11. 11cycliccyclic
  12. 12i=j theoremi=j theorem
  13. 13Fundamental Theorem of cyclic groupsFundamental Theorem of cyclic groups
  14. 14Permutation and Permutation GroupPermutation and Permutation Group
  15. 15Theorem: Disjoint cycles ________commute
  16. 16The order of a permutation is the...LCM of the lengths of its disjoint cycles
  17. 17Every element of S_n can be written asa product of transpositions (2 cycles)
  18. 18even permutationseven permutations
  19. 19alternating group (A_n)alternating group (A_n)
  20. 20isomorphismisomorphism
  21. 21Cayley's TheoremCayley's Theorem
  22. 22AutomorphismAutomorphism
  23. 23Thm: Aut(G) and Inn(G) are...Thm: Aut(G) and Inn(G) are...
  24. 24Thm: Aut(Z_n) is isomorphic toU(n)
  25. 25CosetsCosets
  26. 26LaGrange TheoremLaGrange Theorem
  27. 27Any group of prime order is...cyclic
  28. 28In a finite group, the order of each element....divides the order of the group
  29. 29orbit and stabilizerorbit and stabilizer
  30. 30orbit-stab thmorbit-stab thm
  31. 31external direct productexternal direct product
  32. 32external direct product theoremexternal direct product theorem
  33. 33normal subgroupnormal subgroup
  34. 34normal subgroup testnormal subgroup test
  35. 35Thm: If H is a normal subgroup of G then the set G/H =Thm: If H is a normal subgroup of G then the set G/H =
  36. 36G/Z theoremG/Z theorem
  37. 37internal direct productinternal direct product
  38. 38If G is a group of order p^2, then either...If G is a group of order p^2, then either...
  39. 39homomorphismhomomorphism
  40. 40kernelkernel
  41. 41—
  42. 42Fundamental Thm of Group HomosFundamental Thm of Group Homos
  43. 43determining isomorphism classesdetermining isomorphism classes
  44. 44ring defring def
  45. 45things about ringsthings about rings
  46. 46subring defsubring def
  47. 47subring test thmsubring test thm
  48. 48zero-divisorzero-divisor
  49. 49integral domain defintegral domain def
  50. 50Euclid's LemmaEuclid's Lemma
  51. 51integral domain is a commutative ring...integral domain is a commutative ring...
  52. 52fieldfield
  53. 53Every finite integral domain is afield
  54. 54Z_p is afield
  55. 55characteristic of a ringcharacteristic of a ring
  56. 56If R is a ring with unity 1...char(R) is the order of
  57. 57char(Z_n)=n
  58. 58The characteristic of an integral domain is eitherzero or prime
  59. 59two-sided idealtwo-sided ideal
  60. 60Ideal testIdeal test
  61. 61factor ringfactor ring
  62. 62prime idealprime ideal
  63. 63A is a prime ideal of R >R/A is an integral domain
  64. 64maximal idealmaximal ideal
  65. 65A is a maximal ideal of R >R/A is a field
  66. 66ring homomorphismring homomorphism
  67. 67If phi is a bijection then phi is aring isomorphism
  68. 68first iso thm for ringsfirst iso thm for rings
  69. 69kernels are idealkernels are ideal
  70. 70Let R be a ring with 1. The map phi Z->R given by phi(n)=n*1is a ring homo
  71. 71corollary: Let R be a ring with 1corollary: Let R be a ring with 1
  72. 72corollary: If f is a field thencorollary: If f is a field then
  73. 73If D is an integral domain thenD[x] is an integral domain
  74. 74polynomial ringR[x]
  75. 75If F is a field is F[x] a field?NO
  76. 76Division algorithm for F[x]Division algorithm for F[x]
  77. 77principal ideal domain & thmprincipal ideal domain & thm
  78. 78Let F be a field and let A be a nonzero ideal if F[x], g(x) exists in F[x]Let F be a field and let A be a nonzero ideal if F[x], g(x) exists in F[x]
  79. 79inducible over Dinducible over D
  80. 80Let F be a field. If f(x) exists in F[x] has degree 2 or 3, thenLet F be a field. If f(x) exists in F[x] has degree 2 or 3, then
  81. 81thm: Let f(x) exists in Z[x]. If f(x) is reducible over Q, then...thm: Let f(x) exists in Z[x]. If f(x) is reducible over Q, then...
  82. 82mod-p irreducibility testmod-p irreducibility test
  83. 83Einstein's CriterionEinstein's Criterion
  84. 84Let p(x) exists in F[x]. Then p is maximal in F[x]....Let p(x) exists in F[x]. Then p is maximal in F[x]....
  85. 85corollaries for irreducibilitycorollaries for irreducibility

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Description
Mathematics Flashcards on Abstract Algebra Review, created by Alyssa Baxter on 05/12/2022.
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Alyssa Baxter
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  • Created by Alyssa Baxter · Dec 5, 2022
  • Copied by Alyssa Baxter · Dec 5, 2022
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