Theory Of Equations

I've always wanted to master theory of equations, since the early 1990s. Didn't really got anywhere, except knowing it is just today's abstract algebra. Since 2000s, i understand, more specifically, it's about Galois theory. But today, Wikipedia gives a more complete picture.

A related aspiration was to understand a proof of Fundamental theorem of algebra. As it was popularly known, Gauss is first to prove it. But actually, according to Wikipedia, not so, and gave quite some detail. Also of interest is this Wikipedia quote:

In spite of its name, there is no known purely algebraic proof of the theorem, and many mathematicians believe that such a proof does not exist.[1]

Besides, it is not fundamental for modern algebra; its name was given at a time in which algebra was basically about solving polynomial equations with real or complex coefficients.

Wikipedia is incredibly fantastic today.

Now, the question is, what are some candidates that'd be considered as most important in algebra?