Root objects are the way that

*Mathematica* represents

*algebraic numbers*. Algebraic numbers have the property that when you perform algebraic operations on them, you always get a single algebraic number as the result.

If

Solve and

ToRadicals do not succeed in expressing the solution to a particular polynomial equation in terms of radicals, then it is a good guess that this fundamentally cannot be done. However, you should realize that there are some special cases in which a reduction to radicals is in principle possible, but

*Mathematica* cannot find it. The simplest example is the equation

*x*^{5}+20*x*+32=0, but here the solution in terms of radicals is very complicated. The equation

*x*^{6}-9*x*^{4}-4*x*^{3}+27*x*^{2}-36*x*-23 is another example, where now

is a solution.

Beyond degree four, most polynomials do not have roots that can be expressed at all in terms of radicals. However, for degree five it turns out that the roots can always be expressed in terms of elliptic or hypergeometric functions. The results, however, are typically much too complicated to be useful in practice.