How did the search for solutions of polynomial equations transform algebra in the 19th century?

The search for solutions to polynomial equations was the primary catalyst for the transformation of algebra in the 19th century, shifting the field from a focus on solving specific equations to the study of abstract structures.12

In the 18th century, mathematicians like Joseph-Louis Lagrange analyzed why radical solutions existed for equations up to the fourth degree but failed for higher degrees. By examining how expressions of roots remained invariant under permutation, Lagrange laid the groundwork for future breakthroughs. This line of inquiry led to the realization that a general radical solution for quintic equations was impossible, a fact rigorously proven by Niels Henrik Abel in 1824.13

This impasse did not end research; instead, it prompted a shift toward understanding the conditions under which equations are solvable. Évariste Galois revolutionized this by introducing the Galois group, which linked the solvability of an equation to the properties of its permutation group. This work established Galois theory, which eventually evolved into the study of abstract group theory and field extensions, forming the foundation of modern algebra.345