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Well, as the title says I would like to know a good and direct book about Galois Theory (and also that apllies the theory for the study of polynomials). It's been kinda impossible for me to find such one, since I have tried a few books (non-intuitive and non-direct, one of them was a Thomas Hungerford's) on this theory.

About the other question: I know that there are no solutions in radicals of higher than 4 degree polynomial equations, however, I have seen these questions: Trigonometric solution to solvable equations and Does this higher-order polynomial have an analytic solution?. So, my question is:

And what about the unsolvable ones? We can only find the roots through numerical methods? Also, which books about this would you recommend me reading?

José Carlos Santos
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Mr. N
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    It is easy to find an approximation of the roots of a complex polynomial (for any initial point, the gradient descent, minimizing $|f(z)|^2$, converges to a root, once you have found an approximation $b$ of a root $a$ you can approximate $f(z)/(z-a) \approx (f(z)-f(b))/(z-b)=g(z)$ and repeat the gradient descent with $g$). The radical/solvable stuffs are about polynomials with closed-form factorization in term of elementary functions. – reuns Dec 19 '19 at 17:20

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Concerning Galois theory, I suggest Jean-Pierre Tignol's Galois' Theory of Algebraic Equations.

And concerning the solution of higher degree equations, I suggest Jerry Shurman's Geometry of the Quintic.

José Carlos Santos
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