Math 2.0

statement theorem

Every polynomial has a root in some extension field

Statement

Let KK be a field and let f∈K[x]f \in K[x] with deg⁡f≥1\deg f \ge 1. Then there is a field extension L/KL/K with [L:K]≤deg⁡f[L : K] \le \deg f such that ff has a root in LL.

Remarks

The proof: let gg be an irreducible factor of ff in K[x]K[x]. Adjoining a root of gg gives the field L=K[x]/(g)L = K[x]/(g) of degree deg⁡g\deg g over KK. The class of xx in LL is a root of gg, and so of ff. The irreducible factor gg exists because K[x]K[x] is a unique factorisation domain. Repeating the step gives a splitting field of ff.

Explanations

L1 human

A polynomial with no root in a field gets a root in a bigger field. The bigger field is built from the polynomial itself, without any reference to the complex numbers.

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What this needs

To read this node, read these 43 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. Unit of a ring definition
  7. Zero divisor definition
  8. Divisibility in a commutative ring definition
  9. Ideal generated by a subset definition
  10. Maximal ideal definition
  11. Two-sided ideal definition
  12. Field definition
  13. Degree of a polynomial definition
  14. Integral domain definition
  15. Principal ideal definition
  16. Associate elements definition
  17. Prime element definition
  18. Quotient ring definition
  19. The only ideals of a field are zero and the whole field statement
  20. Irreducible element definition
  21. Field of fractions definition
  22. Leading coefficient definition
  23. Euclidean domain definition
  24. Correspondence theorem for rings statement
  25. Greatest common divisor in an integral domain definition
  26. Principal ideal domain definition
  27. Irreducible polynomial definition
  28. Unique factorisation domain definition
  29. Degree of a product statement
  30. Monic polynomial definition
  31. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  32. Content of a polynomial definition
  33. R/M is a field iff M is maximal statement
  34. Division algorithm for polynomials statement
  35. R[x] is an integral domain iff R is statement
  36. Primitive polynomial definition
  37. K[x] is a Euclidean domain statement
  38. K[x]/(f) is a field iff f is irreducible statement
  39. Product of primitive polynomials is primitive statement
  40. Adjoining a root of a polynomial definition
  41. Gauss's lemma statement
  42. Polynomial ring over a UFD is a UFD statement
  43. Every polynomial has a root in some extension field statement

What depends on this

3 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Splitting_field, Constructing splitting fields

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