RootReduce[expr] attempts to reduce expr to a single Root object.
If expr consists only of integers and Root objects combined using algebraic operations, then the result from RootReduce[expr] will always be a single Root object.
Simple Root objects may in turn automatically evaluate to rational expressions or combinations of radicals.
See The Mathematica Book: Section 3.4.3.
See also: FullSimplify, Solve, ToRadicals.
Related package: NumberTheory`PrimitiveElement`.