Galois is known for his contributions to group theory. He produced a method of determining when a general equation could be solved by radicals. He jotted down his theory the night before his duel and his paper was sent to his friend Auguste Chevalier. Joseph Liouville published his work in 1846. In 1824 Abel gave the first accepted proof of the insolubility of the quintic.
Abelian group is a crucial element in the modern treatment of the Fermat problem.
It is a group where the order of mathematical operations can be reversed without affecting the outcome. Dedekind introduced the notion of an ideal which is fundamental to ring theory. Subsequently, ideals were destined to inspire Barry Mazur, and Andrew Wiles would utilize Mazur's work. He elevated the status of topology with his publication of Analysis Situs . He studied periodic functions in the complex plane.