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Algebraics

represents the domain of algebraic numbers, as in x∈Algebraics.

Details
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Examples  
Basic Examples  
Scope  
Properties & Relations  
Possible Issues  
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Algebraics

Algebraics

represents the domain of algebraic numbers, as in x∈Algebraics.

Details

  • Algebraic numbers are defined to be numbers that solve polynomial equations with rational coefficients.
  • x∈Algebraics evaluates immediately only for quantities x that are explicitly constructed from rational numbers, radicals, and Root objects, or are known to be transcendental.
  • Simplify[expr∈Algebraics] can be used to try to determine whether an expression corresponds to an algebraic number.
  • Algebraics is output in TraditionalForm as TemplateBox[{}, Algebraics]. This typeset form can be input using algs.

Examples

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Basic Examples  (4)

An algebraic number:

Wolfram Language code: Element[(6 + I Sqrt[5]) ^ (1 / 3), Algebraics]

is not an algebraic number:

Wolfram Language code: Element[Pi, Algebraics]

The square root of an algebraic number is an algebraic number:

Wolfram Language code: Simplify[Element[Sqrt[x], Algebraics], Element[x, Algebraics]]

Find algebraic solutions of an equation:

Wolfram Language code: Reduce[(x ^ 2 - 2)(x ^ 2 - E) == 0, x, Algebraics]

Scope  (4)

Test domain membership of a numeric expression:

Wolfram Language code: Element[Sqrt[2] ^ Sqrt[2], Algebraics]
Wolfram Language code: Element[Exp[Pi], Algebraics]

Make domain membership assumptions:

Wolfram Language code: Refine[Element[Sqrt[x] ^ 2 + y ^ (2 / 3), Algebraics], Element[x | y, Algebraics]]

Specify the default domain for Reduce and Resolve:

Wolfram Language code: Reduce[(x ^ 3 - 7)(x ^ 2 - Pi ^ 2) == 0, x, Algebraics]
Wolfram Language code: Resolve[Exists[{x, y}, 1 / 2 <= x ^ 2 + y ^ 2 <= 1 && y == x ^ 2], Algebraics]

TraditionalForm of formatting:

Wolfram Language code: Algebraics//TraditionalForm

Properties & Relations  (3)

Algebraics contains Rationals, Integers, and Primes:

Wolfram Language code: Refine[Element[x, Algebraics], Element[x, #]]& /@ {Rationals, Integers, Primes}

Algebraics is contained in Complexes:

Wolfram Language code: Refine[Element[x, Complexes], Element[x, Algebraics]]

Algebraics neither contains nor is contained in Reals:

Wolfram Language code: {Element[#, Reals], Element[#, Algebraics]}& /@ {Pi, I}

Possible Issues  (1)

Some numbers are not yet known to be algebraic or not:

Wolfram Language code: EulerGamma ∈ Algebraics
Wolfram Research (1999), Algebraics, Wolfram Language function, https://reference.wolfram.com/language/ref/Algebraics.html (updated 2017).

Text

Wolfram Research (1999), Algebraics, Wolfram Language function, https://reference.wolfram.com/language/ref/Algebraics.html (updated 2017).

CMS

Wolfram Language. 1999. "Algebraics." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/Algebraics.html.

APA

Wolfram Language. (1999). Algebraics. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Algebraics.html

BibTeX

@misc{reference.wolfram_2026_algebraics, author="Wolfram Research", title="{Algebraics}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/Algebraics.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_algebraics, organization={Wolfram Research}, title={Algebraics}, year={2017}, url={https://reference.wolfram.com/language/ref/Algebraics.html}, note=[Accessed: 13-June-2026]}