Abel–Ruffini theorem

The Abel–Ruffini theorem states that no formula using arithmetic operations and radicals solves every polynomial equation of degree five or higher with arbitrary coefficients.

Galois theory: The study of field symmetries and their connections to polynomial equations and their roots. It translates solvability by radicals into a question about permutation groups.
Polynomial: An expression built from variables and coefficients using addition, subtraction, multiplication, and nonnegative integer powers. The theorem concerns roots of polynomial equations, not arbitrary equations.
Paolo Ruffini: An Italian mathematician who published an early proof that the general quintic is not solvable by radicals. His 1799 and later arguments anticipated the theorem, though they were not fully accepted.
Bring radical: A special function that expresses a solution of certain quintic equations using an infinite series. It solves some quintics through a broader function class than radicals.
Solvable group: A group whose derived series eventually reaches the identity subgroup. A polynomial is solvable by radicals when its Galois group is solvable.
Radical: An expression involving a root of an indicated degree, such as a square root or cube root. Radicals are the permitted root-extraction operations in the theorem’s claim.
Niels Henrik Abel: A Norwegian mathematician whose proof established the impossibility of a general radical solution for the quintic. His 1824 proof gave the theorem its lasting foundation.
Casus irreducibilis: A case in which an irreducible cubic has three real roots but radical formulas require complex intermediate values. It shows that even solvable equations can demand unexpected intermediates in radical expressions.
Galois group: The group of field automorphisms that preserve a polynomial’s coefficients while permuting its roots. The general quintic’s Galois group fails the condition required for radical solutions.
Polynomial root: A value that makes a polynomial equal to zero. The theorem concerns whether all roots can be expressed through a fixed radical formula.

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