Math 2.0

Learn Year 2

Rings and fields

180 ideas in teaching order. Each idea comes after the ideas that it builds on. 179 of them have a page so far; the others are on the list in grey.

Start with Ring →

  1. Ring definition
  2. Commutative ring definition
  3. The ring of integers example
  4. Zero ring example
  5. Product of rings definition
  6. Subring definition
  7. Unit of a ring definition
  8. Group of units of a ring definition
  9. Zero divisor definition
  10. Nilpotent element definition
  11. Idempotent element definition
  12. Integral domain definition
  13. Cancellation in an integral domain statement
  14. Field definition
  15. A finite integral domain is a field statement
  16. Z/pZ is a field for p prime statement
  17. Division ring definition
  18. The quaternions example
  19. Wedderburn's little theorem statement
  20. Characteristic of a ring definition
  21. The characteristic of an integral domain is zero or prime statement
  22. Prime subfield definition
  23. Freshman's dream statement
  24. Frobenius endomorphism definition
  25. Ring homomorphism definition
  26. Ring isomorphism definition
  27. Kernel of a ring homomorphism definition
  28. A ring homomorphism is injective iff its kernel is zero statement
  29. Ring homomorphisms from a field are injective statement
  30. Ideal definition
  31. Two-sided ideal definition
  32. Ideal generated by a subset definition
  33. Principal ideal definition
  34. sum concept
  35. The only ideals of a field are zero and the whole field statement
  36. Quotient ring definition
  37. Universal property of the quotient ring statement
  38. First isomorphism theorem for rings statement
  39. Second isomorphism theorem for rings statement
  40. Third isomorphism theorem for rings statement
  41. Correspondence theorem for rings statement
  42. Prime ideal definition
  43. Maximal ideal definition
  44. R/P is an integral domain iff P is prime statement
  45. R/M is a field iff M is maximal statement
  46. Maximal ideals are prime statement
  47. Existence of maximal ideals statement
  48. Coprime ideals definition
  49. Chinese remainder theorem for rings statement
  50. Field of fractions definition
  51. The rationals as the field of fractions of the integers example
  52. Polynomial ring definition
  53. Degree of a polynomial definition
  54. Leading coefficient definition
  55. Monic polynomial definition
  56. Degree of a product statement
  57. R[x] is an integral domain iff R is statement
  58. Polynomial evaluation definition
  59. Evaluation homomorphism definition
  60. Universal property of the polynomial ring statement
  61. Polynomials in several variables definition
  62. Division algorithm for polynomials statement
  63. Root of a polynomial definition
  64. Remainder theorem statement
  65. Factor theorem statement
  66. Multiplicity of a root definition
  67. A polynomial over a field has at most as many roots as its degree statement
  68. Formal derivative definition
  69. Repeated roots and the derivative statement
  70. Separable polynomial definition
  71. K[x] is a Euclidean domain statement
  72. Divisibility in a commutative ring definition
  73. Associate elements definition
  74. Irreducible element definition
  75. Prime element definition
  76. Prime elements are irreducible in an integral domain statement
  77. Irreducible elements need not be prime counterexample
  78. Greatest common divisor in an integral domain definition
  79. Euclidean domain definition
  80. Euclidean algorithm in a Euclidean domain concept
  81. Bézout's identity in a Euclidean domain statement
  82. The integers are a Euclidean domain statement
  83. Gaussian integers object
  84. Norm on the Gaussian integers definition
  85. Units of the Gaussian integers statement
  86. The Gaussian integers are a Euclidean domain statement
  87. Principal ideal domain definition
  88. Euclidean domains are principal ideal domains statement
  89. Z[x] is not a principal ideal domain counterexample
  90. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  91. Unique factorisation domain definition
  92. In a UFD irreducible elements are prime statement
  93. PID implies UFD statement
  94. Z[√-5] is not a UFD counterexample
  95. Content of a polynomial definition
  96. Primitive polynomial definition
  97. Product of primitive polynomials is primitive statement
  98. Gauss's lemma statement
  99. Polynomial ring over a UFD is a UFD statement
  100. Irreducible polynomial definition
  101. Polynomials of degree two or three are irreducible iff they have no roots statement
  102. Rational root theorem statement
  103. Eisenstein's criterion statement
  104. Reduction modulo p test for irreducibility statement
  105. Cyclotomic polynomial definition
  106. Irreducibility of the cyclotomic polynomials over the rationals statement
  107. K[x]/(f) is a field iff f is irreducible statement
  108. Ascending chain condition definition
  109. Noetherian ring definition
  110. Hilbert's basis theorem statement
  111. Subfield definition
  112. Field extension definition
  113. Intermediate field definition
  114. Degree of a field extension definition
  115. Finite extension definition
  116. Tower law statement
  117. Algebraic element definition
  118. Transcendental element definition
  119. Minimal polynomial of an algebraic element definition
  120. The minimal polynomial is irreducible statement
  121. The minimal polynomial divides every polynomial vanishing at the element statement
  122. Field generated by adjoining elements definition
  123. Simple extension definition
  124. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  125. Degree of a simple extension is the degree of the minimal polynomial statement
  126. Algebraic extension definition
  127. Finite extensions are algebraic statement
  128. An algebraic extension of an algebraic extension is algebraic statement
  129. The algebraic numbers form a field statement
  130. Adjoining a root of a polynomial definition
  131. Every polynomial has a root in some extension field statement
  132. Split polynomial definition
  133. Splitting field definition
  134. Existence of splitting fields statement
  135. Uniqueness of splitting fields statement
  136. Algebraically closed field definition
  137. Algebraic closure definition
  138. Existence of an algebraic closure statement
  139. Uniqueness of the algebraic closure statement
  140. Finite field definition
  141. The order of a finite field is a prime power statement
  142. Every element of a finite field of order q satisfies x^q = x statement
  143. Finite subgroups of the multiplicative group of a field are cyclic statement
  144. The multiplicative group of a finite field is cyclic statement
  145. Primitive element of a finite field definition
  146. A finite field of order q is the splitting field of x^q - x statement
  147. Galois field object
  148. Existence of finite fields of every prime power order statement
  149. Uniqueness of finite fields of a given order statement
  150. Frobenius automorphism of a finite field definition
  151. Subfields of a finite field statement
  152. Straightedge and compass construction concept
  153. Constructible number definition
  154. The constructible numbers form a field closed under square roots statement
  155. The degree of a constructible number is a power of two statement
  156. Doubling the cube problem
  157. Trisecting the angle problem
  158. Squaring the circle problem
  159. Impossibility of doubling the cube statement
  160. Impossibility of trisecting the angle statement
  161. Transcendence of pi statement
  162. Impossibility of squaring the circle statement
  163. Gauss–Wantzel theorem statement
  164. Symmetric polynomial definition
  165. Elementary symmetric polynomial definition
  166. Fundamental theorem of symmetric polynomials statement
  167. Power sum symmetric polynomial definition
  168. Newton's identities statement
  169. Vieta's formulas statement
  170. Discriminant of a polynomial definition
  171. Galois group definition
  172. Galois group of a polynomial definition
  173. The Galois group has order at most the degree statement
  174. Fixed field definition
  175. Normal extension definition
  176. Separable extension definition
  177. Galois extension definition
  178. Order of the Galois group of a Galois extension equals its degree statement
  179. Primitive element theorem statement
  180. Fundamental theorem of Galois theory statement