WOLFRAM

Wolfram Language & System Documentation Center

RootReduce[expr]

attempts to reduce expr to a single Root object.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Method  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
Cite this Page

RootReduce

RootReduce[expr]

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.

Examples

open all close all

Basic Examples  (1)

Reduce to a single Root object:

Wolfram Language code: RootReduce[Sqrt[2] + Sqrt[3]]

Scope  (2)

Combinations of radical expressions:

Wolfram Language code: RootReduce[2 ^ (1 / 3) + Sqrt[5]]

Combinations of Root objects:

Wolfram Language code: RootReduce[Root[# ^ 5 + 11# + 1&, 1]Root[# ^ 3 + # + 17&, 1]]

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

Wolfram Language code: RootReduce[Sqrt[Root[# ^ 3 - 7# - 2&, 1] + 1] / AlgebraicNumber[3 ^ (1 / 3), {1, 2, 3}]]

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

Wolfram Language code: RootReduce[Sqrt[7] / Sqrt[5 + 2Sqrt[6]]]
Wolfram Language code: RootReduce[(Sqrt[2] + Sqrt[3] + Sqrt[6] + 3) / Sqrt[5 + 2Sqrt[6]]]
Wolfram Language code: RootReduce[(Sqrt[18] + Sqrt[27]) / Sqrt[5 + 2Sqrt[6]]]

Options  (1)

Method  (1)

By default, RootReduce heuristically selects the method to use:

Wolfram Language code: a = 2 2 ^ (1 / 3) + 3 3 ^ (1 / 7) + 5 2 ^ (1 / 3)3 ^ (1 / 7); RootReduce[a]//Timing

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

Wolfram Language code: RootReduce[a, Method -> "NumberField"]//Timing

The other available method recursively performs arithmetic operations:

Wolfram Language code: RootReduce[a, Method -> "Recursive"]//Timing

Here the "Recursive" method is faster:

Wolfram Language code: b = RootReduce[2 ^ (1 / 3) + 3 ^ (1 / 3) + 1]; c = b - 2 ^ (1 / 3) - 3 ^ (1 / 3); RootReduce[c]//Timing
Wolfram Language code: RootReduce[c, Method -> "NumberField"]//Timing
Wolfram Language code: RootReduce[c, Method -> "Recursive"]//Timing

Applications  (1)

The numeric test used by Equal cannot prove the equality:

Wolfram Language code: Sqrt[2] + Sqrt[3] + Sqrt[5] == Sqrt[10 + 2Sqrt[15] + 4Sqrt[4 + Sqrt[15]]]

RootReduce proves that the two algebraic numbers are equal:

Wolfram Language code: RootReduce[%]

FullSimplify will use RootReduce:

Wolfram Language code: FullSimplify[%%]

Properties & Relations  (3)

The results given by RootReduce are canonical:

Wolfram Language code: algs = {Sqrt[2] + Sqrt[3] + Sqrt[5], Sqrt[10 + 2Sqrt[15] + 4Sqrt[4 + Sqrt[15]]], Sqrt[25 + 4Sqrt[15] + 2Sqrt[46 + 8Sqrt[15]]] - Sqrt[5]}
Wolfram Language code: RootReduce /@ algs

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

Wolfram Language code: a = RootReduce[Root[# ^ 3 + # + 11&, 1] + Root[# ^ 3 + # + 17&, 1]]
Wolfram Language code: MinimalPolynomial[a, x]

In exceptional cases the result can have a lower degree:

Wolfram Language code: b = RootReduce[a - Root[# ^ 3 + # + 11&, 1]]
Wolfram Language code: MinimalPolynomial[b, x]

Root objects can be converted to AlgebraicNumber objects:

Wolfram Language code: ToNumberField[Root[# ^ 5 + # + 11&, 1]]

RootReduce converts from AlgebraicNumber objects:

Wolfram Language code: RootReduce[%]
Wolfram Research (1996), RootReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/RootReduce.html (updated 2007).

Text

Wolfram Research (1996), RootReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/RootReduce.html (updated 2007).

CMS

Wolfram Language. 1996. "RootReduce." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/RootReduce.html.

APA

Wolfram Language. (1996). RootReduce. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RootReduce.html

BibTeX

@misc{reference.wolfram_2026_rootreduce, author="Wolfram Research", title="{RootReduce}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RootReduce.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_rootreduce, organization={Wolfram Research}, title={RootReduce}, year={2007}, url={https://reference.wolfram.com/language/ref/RootReduce.html}, note=[Accessed: 13-June-2026]}