TheoremProved

Abel–Ruffini theorem

Statement

There is no general formula, using only addition, subtraction, multiplication, division and radicals (roots) of the coefficients, that expresses the roots of every polynomial equation of degree $5$ or higher.

Why is it true?

Degree $2,3,4$ equations can always be cracked open by a fixed recipe of arithmetic and roots — the quadratic formula and its degree-3 and degree-4 cousins. Abel and Ruffini showed that starting at degree $5$ , this recipe-based approach runs out: the symmetry group that governs how the roots can be permuted becomes too complicated (it is no longer solvable in the group-theoretic sense) for any tower of root-extractions to unscramble it. Individual quintics can still be solved with radicals — it is the general formula that is impossible.

Proof sketch

Galois theory recasts solvability by radicals as a property of the Galois group of the polynomial: the roots can be expressed by radicals exactly when this group is solvable (built up from abelian pieces). For a generic degree- $5$ polynomial the Galois group is the full symmetric group $S_5$ , and $S_5$ is not solvable because its only proper nontrivial normal subgroup, $A_5$ , is simple and non-abelian; hence no radical formula can exist for the general quintic.

Proved by

proved 1824Niels Henrik Abel

Topics that use this theorem

Galois theory

Related theorems

Fundamental theorem of Galois theory Lagrange's theorem (group theory)

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Ian Stewart (2015). Galois Theory · DOI:10.1201/b18187