Here, however, is where his genius truly took off. Galois managed to associate with each equation a sort of “genetic code” of that equation-the Galois group of the equation-and to demonstrate that the properties of the Galois group determine whether the equation is solvable by a formula or not. Symmetry became the key concept, and the Galois group was a direct measure of the symmetry properties of an equation.
-Mario Livio
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