# Algebraic Numbers

Root[f,k] | the k root of the polynomial equation f[x]==0 |

The representation of algebraic numbers.

Root objects are the way that the Wolfram Language 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.

RootReduce[expr] | attempt to reduce expr to a single Root object |

ToRadicals[expr] | attempt to transform Root objects to explicit radicals |

Operations on algebraic numbers.

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 the Wolfram System cannot find it. The simplest example is the equation , but here the solution in terms of radicals is very complicated. The equation 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.

RootSum[f,form] | the sum of form[x] for all satisfying the polynomial equation f[x]==0 |

Normal[expr] | the form of expr with RootSum replaced by explicit sums of Root objects |

RootApproximant[x] | converts the number x to one of the "simplest" algebraic numbers that approximates it well |

RootApproximant[x,n] | finds an algebraic number of degree at most n that approximates x |