As Evariste Galois noted above, algebra in the 19th century was in dire need of a major revolution, but like many revolutionaries he found himself facing an endless series of road blocks and block heads, and it was not until many years after his tragic death that his ideas came to be understood, appreciated and applied in a vast number of areas of mathematics and science.

One of the great uses of algebra from ancient times was the solution of polynomial equations. Increasingly complicated formulae were found for the solution of quadratic, cubic and quartic equations, that is general equations in a single unknown taken to the powers of 2,3, and 4. But no one was able to solve the general quintic, or the polynomial that took the unknown to the 5th power:

Galois looked at the family of quintics to determine which ones can in fact be solved, and he showed how his procedures could be generalized to explore the solvability of polynomials of any degree.

Galois achieved this breakthrough by employing a new algebraic structure called a group. Groups had been explored before in some isolated cases, but Galois was the first to show the true power of group theory.

Galois: Biography of a Great Thinker

What is a Galois group and how is it used?

What does Galois theory tell us about equations of degree five and higher?

Evariste Galois a documentary

Abstract algebra and group theory seemed too abstruse and rarified to be of any practical value for science, as the great astronomer James Jeans noted:

But it turned out that group theory could be applied to every scientific field that dealt in some way with symmetry, which is pretty much all of science!