Math 2.0

Field theory and polynomials Galois theory Theorem

Abel–Ruffini theorem

Builds on Solvable by radicalsSymmetric groupSolvable groupThe symmetric group on five or more letters is not solvableGalois' theorem on solvability

Statement

Let n≥5n \ge 5 be an integer.

  1. Let t1,…,tnt_1, \dots, t_n be indeterminates and K=ℚ(t1,…,tn)K = \Q(t_1, \dots, t_n). The general polynomial xn+t1xn−1+⋯+tn−1x+tn∈K[x]x^n + t_1 x^{n-1} + \dots + t_{n-1} x + t_n \in K[x] is not solvable by radicals over KK.
  2. There is a polynomial of degree nn with rational coefficients that is not solvable by radicals over ℚ\Q.

Remarks

By Galois' theorem on solvability, it suffices to find a Galois group that is not solvable. The Galois group of the general polynomial of degree nn is the full symmetric group SnS_n, and SnS_n is not solvable for n≥5n \ge 5. For clause 2, the polynomial x5−4x+2x^5 - 4x + 2 has Galois group S5S_5 over ℚ\Q; see the quintic equation. Multiplying it by xn−5x^{n-5} gives a polynomial of degree nn with the same nonsolvable roots. The bound n≥5n \ge 5 is necessary: every polynomial of degree at most 44 is solvable by radicals, by the cubic formula and the quartic formula.

Going deeper

For a student

The theorem does not say that quintic equations have no solutions: by the fundamental theorem of algebra, each has five complex roots. It says only that these roots are not always radical expressions in the coefficients. Many quintics, such as x5−2x^5 - 2, are still solvable by radicals; Galois theory decides which ones. The roots can be expressed with other functions, for instance with the Bring radical or with elliptic modular functions.

Reading path

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

  1. Binary operation definition
  2. Characteristic of a field concept
  3. Cycle notation concept
  4. Group concept
  5. Radical extension concept
  6. Solvable by radicals concept
  7. Commutativity definition
  8. Commutator of two elements definition
  9. Conjugate elements definition
  10. Group homomorphism definition
  11. Ring definition
  12. Subgroup definition
  13. Symmetric group definition
  14. Abelian group definition
  15. Kernel of a homomorphism definition
  16. Conjugate subgroup definition
  17. Commutative ring definition
  18. Disjoint cycles commute statement
  19. Polynomial ring definition
  20. Ring homomorphism definition
  21. Subring definition
  22. Transposition definition
  23. Unit of a ring definition
  24. Left coset definition
  25. Right coset definition
  26. Subgroup criterion statement
  27. Normal subgroup definition
  28. Decomposition of a permutation into disjoint cycles statement
  29. Polynomial evaluation definition
  30. Field definition
  31. Ring isomorphism definition
  32. Intersection of subgroups is a subgroup statement
  33. Root of a polynomial definition
  34. Decomposition of a permutation into transpositions statement
  35. Subfield definition
  36. Subgroup generated by a subset definition
  37. Normality via left and right cosets statement
  38. Sign of a permutation definition
  39. Commutator subgroup definition
  40. Field extension definition
  41. Split polynomial definition
  42. Quotient group definition
  43. Even and odd permutations definition
  44. Sign of a cycle statement
  45. Derived series definition
  46. Sign is a homomorphism statement
  47. Intermediate field definition
  48. Subnormal series definition
  49. Galois group definition
  50. Alternating group definition
  51. Solvable group definition
  52. Field generated by adjoining elements definition
  53. The alternating group is generated by three-cycles statement
  54. Splitting field definition
  55. Subgroups of solvable groups are solvable statement
  56. Simplicity of the alternating group statement
  57. Galois group of a polynomial definition
  58. The symmetric group on five or more letters is not solvable statement
  59. Galois' theorem on solvability statement
  60. Abel–Ruffini theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Abel–Ruffini_theorem
About this page
Tag163A2WQQJX
URIm2://163A2WQQJX
Kindstatement, theorem
WritingW0 stub
MSC12F10, 12E12
StatusThis copy of the site has no build data. Run m2 status 163A2WQQJX.
AboutSolvable by radicals
UsesSolvable by radicals, Symmetric group, Solvable group, The symmetric group on five or more letters is not solvable, Galois' theorem on solvability
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Changed2 November 2026

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