Math 2.0

statement theorem

Uniqueness of the algebraic closure

Statement

Let KK be a field, and let LL and L′L' be algebraic closures of KK. Then there is a ring isomorphism σ ⁣:L→L′\sigma \colon L \to L' with σ(c)=c\sigma(c) = c for every c∈Kc \in K.

Remarks

The proof uses Zorn's lemma on the partial isomorphisms between subfields of LL and L′L' that fix KK. The isomorphism is not unique in general. For example, complex conjugation and the identity are two such isomorphisms from C\mathbb{C} to C\mathbb{C} over R\mathbb{R}. So one often writes "the" algebraic closure K‾\overline{K}. The statement is an infinite form of the uniqueness of splitting fields.

Explanations

L1 human

Two algebraic closures of the same field are the same field with different names for the elements. There can be many ways to match them, so the matching is not part of the structure.

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What this needs

To read this node, read these 56 nodes in this order. The foundations come first, and the list ends with this node.

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

Sources

  1. Web https://en.wikipedia.org/wiki/Algebraic_closure, lead section

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