quality:corruption, information:extraction
Permutation group approach

Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say A and B, that A + 5 B = 7. The central idea of Galois' theory is to consider permutations (or rearrangements) of the roots such that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers. It extends naturally to equations with coefficients in any field, but this will not be considered in the simple examples below.

These permutations together form a permutation group, also called the Galois group of the polynomial, which is explicitly described in the following examples.