Galois's Theorem

Galois's theorem states that a polynomial equation over a field of field characteristic zero is solvable by radicals iff the Galois group of its splitting field over is a solvable group (Artin 1971, pp. 72-74). Here solvability by radicals means that all the roots can be expressed from elements of using finitely many additions, subtractions, multiplications, divisions, and extractions of radicals.

For an irreducible polynomial of prime polynomial degree, an equivalent criterion is that every root is a rational function of any two distinct roots, with coefficients in (Weintraub 2009, pp. 99-100). Equivalently, any such pair , generates the splitting field .

The criterion for solvability and the computation of explicit radical expressions are separate tasks. Hanrot and Morain (2001) give algorithms for constructing these expressions for equations with Galois groups that are solvable.