Évariste Galois

Ah! to die so young, and to die for so little!

I have not time. I have not time.

Please make public this letter. I beg you, publicly, to ask Jacobi or Gauss to give their opinion, not on the truth, but on 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.

You know, my dear Auguste, that these are not the only subjects I have explored. For some time, my principal meditations have been directed to the application of the theory of ambiguity to transcendental analysis.

I have made some new discoveries in analysis. Some are on the theory of equations, others on integral functions.

I have no time to write out this long theory and its applications. I have no time. I have no time.

I have shown that the general equation of degree n is solvable by radicals if and only if its Galois group is solvable.

A group is a collection of substitutions such that the product of any two substitutions is also in the collection, and the inverse of any substitution is also in the collection.

The theory of groups is, so to speak, the essence of the theory of equations.

We will call a group primitive if it contains no invariant subgroup other than the identity and the group itself.

The order of a group is the number of distinct substitutions it contains.

The roots of an equation are permuted among themselves by the substitutions of its Galois group.

The properties of a group are independent of the particular representation chosen for its elements.

The theory of groups is a vast and fertile field, whose exploration has only just begun.

The true object of the theory of equations is to find the conditions under which an equation can be solved by radicals.

The concept of a group is fundamental to understanding the structure of algebraic equations.

The theory of groups provides a powerful tool for analyzing the symmetries of mathematical objects.

The conditions for solvability by radicals are intimately connected with the structure of the Galois group.

The problem of solving equations by radicals is reduced to the problem of determining the structure of their Galois groups.

The theory of groups allows us to understand why certain equations cannot be solved by radicals.