Impossibility of squaring the circle

Reading path

To understand this page from the ground up, read these 60 pages in order. The foundations come first.

1. Group concept
2. Straightedge and compass construction concept
3. Ring definition
4. Constructible number definition
5. Squaring the circle 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. Field extension definition
40. R/M is a field iff M is maximal statement
41. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
42. Division algorithm for polynomials statement
43. Algebraic element definition
44. R[x] is an integral domain iff R is statement
45. First isomorphism theorem for rings statement
46. Intermediate field definition
47. Degree of a field extension definition
48. Transcendental element definition
49. Minimal polynomial of an algebraic element definition
50. K[x] is a Euclidean domain statement
51. Field generated by adjoining elements definition
52. Tower law statement
53. Transcendence of pi statement
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. Simple extension definition
57. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
58. Degree of a simple extension is the degree of the minimal polynomial statement
59. The degree of a constructible number is a power of two statement
60. Impossibility of squaring the circle statement