Math 2.0

Field theory and polynomials Theorem

Impossibility of trisecting the angle

Builds on Trisecting the angleThe degree of a constructible number is a power of twoPolynomials of degree two or three are irreducible iff they have no rootsRational root theorem

Statement

The angle π/3\pi/3, that is, 60∘60^\circ, cannot be trisected with straightedge and compass. Equivalently, cos⁡(π/9)\cos(\pi/9) is not a constructible number.

Remarks

The proof: cos⁡3θ=4cos⁡3θ−3cos⁡θ\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta shows that c=cos⁡(π/9)c = \cos(\pi/9) is a root of 8x3−6x−18x^3 - 6x - 1. Then u=2cu = 2c is a root of x3−3x−1x^3 - 3x - 1. The polynomial x3−3x−1x^3 - 3x - 1 has no rational root, by the rational root theorem. So it is irreducible, and [ℚ(c):ℚ]=3[\mathbb{Q}(c) : \mathbb{Q}] = 3. Since 33 is not a power of 22, the number cc is not constructible. The angle π/3\pi/3 itself is constructible, so no general trisection method exists.

History

Pierre Wantzel published the proof in 1837.

Reading path

To understand this page from the ground up, read these 62 pages in order. The foundations come first.

  1. Group concept
  2. Straightedge and compass construction concept
  3. Ring definition
  4. Constructible number definition
  5. Trisecting the angle 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. The only ideals of a field are zero and the whole field statement
  26. Evaluation homomorphism definition
  27. Leading coefficient definition
  28. Root of a polynomial definition
  29. Subfield definition
  30. Irreducible element definition
  31. Euclidean domain definition
  32. Correspondence theorem for rings statement
  33. Principal ideal domain definition
  34. The constructible numbers form a field closed under square roots statement
  35. Monic polynomial definition
  36. Universal property of the quotient ring statement
  37. Irreducible polynomial definition
  38. Degree of a product statement
  39. Rational root theorem statement
  40. Field extension definition
  41. R/M is a field iff M is maximal statement
  42. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  43. Division algorithm for polynomials statement
  44. Algebraic element definition
  45. R[x] is an integral domain iff R is statement
  46. First isomorphism theorem for rings statement
  47. Intermediate field definition
  48. Degree of a field extension definition
  49. Minimal polynomial of an algebraic element definition
  50. Remainder theorem statement
  51. K[x] is a Euclidean domain statement
  52. Tower law statement
  53. Field generated by adjoining elements definition
  54. The minimal polynomial divides every polynomial vanishing at the element statement
  55. K[x]/(f) is a field iff f is irreducible statement
  56. Factor theorem statement
  57. Simple extension definition
  58. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  59. Polynomials of degree two or three are irreducible iff they have no roots statement
  60. Degree of a simple extension is the degree of the minimal polynomial statement
  61. The degree of a constructible number is a power of two statement
  62. Impossibility of trisecting the angle statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Angle_trisection, Proof of impossibility
About this page
Tag0EWXPZCSZ0
URIm2://0EWXPZCSZ0
Kindstatement, theorem
WritingW0 stub
MSC12F05
StatusThis copy of the site has no build data. Run m2 status 0EWXPZCSZ0.
UsesTrisecting the angle, The degree of a constructible number is a power of two, Polynomials of degree two or three are irreducible iff they have no roots, Rational root theorem

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