Higher Degree Polynomials Using Bring Radicals

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It is not possible to solve any quintic with the 4 arithmetic operations and radicals. However, there is a method using Bring Radicals, the solutions to $x^5+x+a$. Is there any known equivalent for higher degree polynomials, and if so, is there a method to generate the reduced polynomials that these new "radicals" would be the solutions for?