Galois's most significant contribution to mathematics by far is his development of Galois theory.

The following proof is based on Galois theory.

The study of such extensions is the starting point of Galois theory.

Initially, his research dealt with Galois theory of equations.

Jordan's work did much to bring Galois theory into the mainstream.

However the original notion in Galois theory is slightly different.

See also differential Galois theory for a more detailed discussion.

The theorem can be proved without any use of Galois theory.

Long standing questions about compass and straightedge construction were finally settled by Galois theory.

In particular he worked on Galois theory, ideals and equations of the fifth degree.