In 1823, Niels Hendrik Abel managed to prove that there is no algebraic formula to solve equations of degree 5pic.twitter.com/3Agja4l6dL
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For any polynomial of degree 4 or less, there is a formula to extract the roots in terms of radicals of the coefficients of the polynomial (e.g. the Quadratic Formula). However, for a general polynomial of degree 5 or higher, there is no such formula.
Since the coefficients of any polynomial are symmetric functions of the roots, it turns out that the solutions form a group structure. And unfortunately to explain this in depth requires a great deal of abstract algebra.
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