An algebraic equation is algebraically solvable iff its group is solvable. In order that an irreducible equation of prime degree be solvable by radicals, it is necessary and sufficient that all its roots be rational functions of two roots.
Galois's Theorem
See also
Abel's Impossibility Theorem, Solvable GroupExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Galois's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaloissTheorem.html