Uniqueness of finite fields of a given order

Reading path

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

1. Group concept
2. Ring definition
3. Characteristic of a ring definition
4. Commutative ring definition
5. Ideal definition
6. Polynomial ring definition
7. Ring homomorphism definition
8. Subring definition
9. Unit of a ring definition
10. Zero divisor definition
11. Divisibility in a commutative ring definition
12. Ideal generated by a subset definition
13. Maximal ideal definition
14. Two-sided ideal definition
15. Degree of a polynomial definition
16. Group of units of a ring definition
17. Polynomial evaluation definition
18. Field definition
19. Integral domain definition
20. Kernel of a ring homomorphism definition
21. Ring isomorphism definition
22. Principal ideal definition
23. Quotient ring definition
24. Leading coefficient definition
25. The characteristic of an integral domain is zero or prime statement
26. The only ideals of a field are zero and the whole field statement
27. Evaluation homomorphism definition
28. Root of a polynomial definition
29. Subfield definition
30. Finite field definition
31. Irreducible element definition
32. Euclidean domain definition
33. Correspondence theorem for rings statement
34. Monic polynomial definition
35. Principal ideal domain definition
36. Split polynomial definition
37. Degree of a product statement
38. Prime subfield definition
39. Universal property of the quotient ring statement
40. Irreducible polynomial definition
41. Every element of a finite field of order q satisfies x^q = x statement
42. Field extension definition
43. Division algorithm for polynomials statement
44. R/M is a field iff M is maximal statement
45. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
46. R[x] is an integral domain iff R is statement
47. Algebraic element definition
48. First isomorphism theorem for rings statement
49. Intermediate field definition
50. Degree of a field extension definition
51. Minimal polynomial of an algebraic element definition
52. K[x] is a Euclidean domain statement
53. The order of a finite field is a prime power statement
54. Field generated by adjoining elements definition
55. The minimal polynomial divides every polynomial vanishing at the element statement
56. K[x]/(f) is a field iff f is irreducible statement
57. Simple extension definition
58. Splitting field definition
59. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
60. A finite field of order q is the splitting field of x^q - x statement
61. Uniqueness of splitting fields statement
62. Uniqueness of finite fields of a given order statement