The theory of solvability of polynomial equations developed by Galois
began with the attempt to find formulas
for the solutions of polynomial equations of degree five.
After the discovery of the fundamental theorem of algebra,
the question of proving the existence of solutions
changed to determining the form of the solutions.
The question was whether or not the solutions could be expressed
in a reasonable way by extracting square roots, cube roots, etc.,
of combinations of the coefficients of the polynomial.
Galois saw that this involved a comparison of two fields,
by determining how the field generated by the coefficients
sits inside the larger field generated by the solutions of the equation.
This would be a good time to read (in succession) all of the chapter introductions in Abstract Algebra. You will also find references to Galois and Galois theory on the web. Here are a few sites that we can recommend:
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