I realized that generally a 5th degree equation cannot be solved by radicals, but what if we have a 5th degree equation with repeated roots, how about its solvability by radicals? Intuitively, it would be like an equation with lower degree while those equations can be solved by radicals; specifically, would there be a 4th degree equation sharing the same roots with the 5th equation with a doube root?
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$\begingroup$Yes: you can find the repeated factor by applying Euclid's algorithm to the polynomial and its derivative, then divide by this factor to get at worst a quartic.
Indeed, here is an even easier method: a repeated root is also a root of the derivative. The derivative is a quartic. So we can find the roots of the derivative, then check which are also roots of the original polynomial. Dividing $x-a$ where $a$ is a repeated root will yield a quartic with the same roots, while dividing by the gcd of the polynomial and its derivative will give a polynomial that is at worst quartic with no repeated roots.
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