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=== Galois theory === {{Main|Galois theory}} Galois's most significant contribution to mathematics is his development of Galois theory. He realized that the algebraic solution to a [[polynomial]] equation is related to the structure of a group of [[permutation]]s associated with the roots of the polynomial, the [[Galois group]] of the polynomial. He found that an equation could be solved in [[nth root|radicals]] if one can find a series of subgroups of its Galois group, each one normal in its successor with [[abelian group|abelian]] quotient, that is, its Galois group is [[solvable group|solvable]]. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the [[theory of equations]] to which Galois originally applied it.<ref name="lie" />
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