The Abel-Ruffini theorem says that there are some fifth-degree equations whose solution cannot be so expressed. The equation \( x^5 - x + 1 = 0 \) is an example. Some other fifth degree equations can be solved by radicals, for example \( x^5 - x^4 - x + 1 = 0 \). The precise criterion that distinguishes between those equations that can be solved by radicals and those that cannot was given by Galois and is now part of Galois theory: a polynomial equation can be solved by radicals if and only if its Galois group is a solvable group.

Today, in the modern algebraic context, we say that second, third and fourth degree polynomial equations can always be solved by radicals because the symmetric groups \( S_2, S_3 \) and \( S_4 \) are solvable groups, whereas \( S_n \) is not solvable for \( n \ge 5 \).