attempts to reduce expr to a single Root object.

Details and Options

  • If expr consists only of integers and Root and AlgebraicNumber 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.
  • RootReduce automatically threads over lists, as well as equations, inequalities, and logic functions.


open allclose all

Basic Examples  (1)

Reduce to a single Root object:

Scope  (2)

Combinations of radical expressions:

Combinations of Root objects:

Reduce any algebraic combination of radicals, Root, and AlgebraicNumber objects:

The result is always a Root object, a quadratic radical expression, or a rational number:

Options  (1)

Method  (1)

By default, RootReduce heuristically selects the method to use:

In this case conversion to AlgebraicNumber objects in a common number field is used:

The other available method recursively performs arithmetic operations:

Here the "Recursive" method is faster:

Applications  (1)

The numeric test used by Equal cannot prove the equality:

RootReduce proves that the two algebraic numbers are equal:

FullSimplify will use RootReduce:

Properties & Relations  (3)

The results given by RootReduce are canonical:

In general the degree of the reduced polynomial will be the product of the degrees:

In exceptional cases the result can have a lower degree:

Root objects can be converted to AlgebraicNumber objects:

RootReduce converts from AlgebraicNumber objects:

Wolfram Research (1996), RootReduce, Wolfram Language function, (updated 2007).


Wolfram Research (1996), RootReduce, Wolfram Language function, (updated 2007).


@misc{reference.wolfram_2021_rootreduce, author="Wolfram Research", title="{RootReduce}", year="2007", howpublished="\url{}", note=[Accessed: 22-October-2021 ]}


@online{reference.wolfram_2021_rootreduce, organization={Wolfram Research}, title={RootReduce}, year={2007}, url={}, note=[Accessed: 22-October-2021 ]}


Wolfram Language. 1996. "RootReduce." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007.


Wolfram Language. (1996). RootReduce. Wolfram Language & System Documentation Center. Retrieved from