From the closing lines of a letter from Galois to his friend Auguste Chevalier, dated May 29, 1832, two days before Galois' death:[23]

Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l'importance des théorèmes.

Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

(Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)

Within the 60 or so pages of Galois' collected works are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.[28][29] His work has been compared to that of Niels Henrik Abel, another mathematician who died at a very young age, and much of their work had significant overlap.

Algebra Edit
While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group (in French groupe) in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as group theory. He developed the concept that is today known as a normal subgroup. He called the decomposition of a group into its left and right cosets a proper decomposition if the left and right cosets coincide, which is what today is known as a normal subgroup.[23] He also introduced the concept of a finite field (also known as a Galois field in his honor), in essentially the same form as it is understood today.[12]

In his last letter to Chevalier[23] and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields:

He constructed the general linear group over a prime field, GL(ν, p) and computed its order, in studying the Galois group of the general equation of degree pν.[30]
He constructed the projective special linear group PSL(2,p). Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3.[31] These were the second family of finite simple groups, after the alternating groups.[32]
He noted the exceptional fact that PSL(2,p) is simple and acts on p points if and only if p is 5, 7, or 11.[33][34]
Galois theory Edit
Main article: Galois theory
Galois' 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 permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, or its Galois group is 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.[28]

Analysis Edit
Galois also made some contributions to the theory of Abelian integrals and continued fractions.

As written in his last letter,[23] Galois passed from the study of elliptic functions to consideration of the integrals of the most general algebraic differentials, today called Abelian integrals. He classified these integrals into three categories.

Continued fractions Edit
In his first paper in 1828,[7] Galois proved that the regular continued fraction which represents a quadratic surd ζ is purely periodic if and only if ζ is a reduced surd, that is, {\displaystyle \zeta >1}\zeta >1 and its conjugate {\displaystyle \eta }\eta satisfies {\displaystyle -1