You're not wrong to feel that way—it's a bit of a running joke in higher algebra! A surprising amount of the development of abstract algebra—especially Galois theory, field theory, and group theory—traces its roots back to the problem of solving polynomial equations by radicals.
Here’s a quick sketch of how central that problem is:
Quadratics to Quartics: Solving polynomials up to degree 4 by radicals was well-established by the Renaissance. This pushed mathematicians to ask: what about degree 5 and higher?
The Quintic Problem: For centuries, people searched in vain for a general formula for solving quintics. This pursuit drove major developments in the theory of equations.
Galois Theory: Évariste Galois reframed the problem in terms of symmetries of the roots of polynomials—what we now call Galois groups. His revolutionary insight was that whether a polynomial is solvable by radicals depends on whether its Galois group is solvable (in the group-theoretic sense).
Birth of Modern Algebra: Galois's approach laid the groundwork for group theory, field theory, and a large chunk of abstract algebra. Concepts like normal subgroups, field extensions, and automorphisms all emerged from this.
So yes, in a way, algebra was built to answer the quintic question—and the answer turned out to require inventing a huge amount of mathematical machinery. That’s partly why the quintic feels like the gravitational center of classical algebra.