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I know the work from Galois showing it is not possible to solve certain equations using only a certain type of numbers. But that was more than 100 years ago, I suppose lots of progress has been made on this topic.

So my question is, What has been discovered to solve polynomial equations? I don't mean algorithmic techniques, I mean theoretical calculations and exact solutions.

He showed some equations are not solvable using specific numbers, Has it been solved using another kind of numbers or something like that?

If I give you a degree 500 polynomial, Do you have the mathematical techniques to solve the equation and find the 500 solutions? I don't mean numerical calculations.