Why quintic polynomial equations are not solvable in radicals

summary:We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations $+,\,-,\,\cdot,\,:\,$, and $\root n \of{\cdot}$. Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals