Answer
Galois' necessary and sufficient condition for a polynomial to be solvable by radicals states that a polynomial is solvable by radicals if and only if its Galois group is solvable. In simpler terms, a polynomial can be solved using a finite sequence of algebraic operations (addition, subtraction, multiplication, division, and taking roots) if and only if its group of symmetries (the Galois group) has a specific structure.