Math 2.0

Field theory and polynomials Theorem

Impossibility of doubling the cube

Builds on Doubling the cubeThe degree of a constructible number is a power of twoEisenstein's criterionDegree of a simple extension is the degree of the minimal polynomial

Statement

The number 23\sqrt[3]{2} is not a constructible number. So the cube cannot be doubled with straightedge and compass.

Remarks

The proof: x3−2x^3 - 2 is irreducible over ℚ\mathbb{Q}, by Eisenstein's criterion with p=2p = 2. So [ℚ(23):ℚ]=3[\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3. Since 33 is not a power of 22, the number 23\sqrt[3]{2} is not constructible, by the degree theorem.

History

Pierre Wantzel proved the impossibility in 1837.

Reading path

To understand this page from the ground up, read these 68 pages in order. The foundations come first.

  1. Group concept
  2. Straightedge and compass construction concept
  3. Ring definition
  4. Constructible number definition
  5. Doubling the cube problem
  6. Commutative ring definition
  7. Ideal definition
  8. Polynomial ring definition
  9. Ring homomorphism definition
  10. Subring definition
  11. Unit of a ring definition
  12. Zero divisor definition
  13. Divisibility in a commutative ring definition
  14. Ideal generated by a subset definition
  15. Maximal ideal definition
  16. Two-sided ideal definition
  17. Degree of a polynomial definition
  18. Polynomial evaluation definition
  19. Field definition
  20. Integral domain definition
  21. Kernel of a ring homomorphism definition
  22. Ring isomorphism definition
  23. Principal ideal definition
  24. Quotient ring definition
  25. Associate elements definition
  26. Prime element definition
  27. The only ideals of a field are zero and the whole field statement
  28. Evaluation homomorphism definition
  29. Leading coefficient definition
  30. Root of a polynomial definition
  31. Subfield definition
  32. Irreducible element definition
  33. Field of fractions definition
  34. Euclidean domain definition
  35. Correspondence theorem for rings statement
  36. Greatest common divisor in an integral domain definition
  37. Principal ideal domain definition
  38. Universal property of the quotient ring statement
  39. The constructible numbers form a field closed under square roots statement
  40. Irreducible polynomial definition
  41. Degree of a product statement
  42. Monic polynomial definition
  43. Unique factorisation domain definition
  44. Field extension definition
  45. R/M is a field iff M is maximal statement
  46. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  47. Division algorithm for polynomials statement
  48. Content of a polynomial definition
  49. R[x] is an integral domain iff R is statement
  50. First isomorphism theorem for rings statement
  51. Algebraic element definition
  52. Intermediate field definition
  53. Degree of a field extension definition
  54. Primitive polynomial definition
  55. K[x] is a Euclidean domain statement
  56. Minimal polynomial of an algebraic element definition
  57. Tower law statement
  58. Field generated by adjoining elements definition
  59. The minimal polynomial divides every polynomial vanishing at the element statement
  60. K[x]/(f) is a field iff f is irreducible statement
  61. Product of primitive polynomials is primitive statement
  62. Simple extension definition
  63. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  64. Gauss's lemma statement
  65. Eisenstein's criterion statement
  66. Degree of a simple extension is the degree of the minimal polynomial statement
  67. The degree of a constructible number is a power of two statement
  68. Impossibility of doubling the cube statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Doubling_the_cube, Proof of impossibility
About this page
Tag0CQ357M0QS
URIm2://0CQ357M0QS
Kindstatement, theorem
WritingW0 stub
MSC12F05
StatusThis copy of the site has no build data. Run m2 status 0CQ357M0QS.
UsesDoubling the cube, The degree of a constructible number is a power of two, Eisenstein's criterion, Degree of a simple extension is the degree of the minimal polynomial

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