# AlgebraicNumber

AlgebraicNumber[θ,{c0,c1,,cn}]

represents the algebraic number in the field given by .

# Details

• AlgebraicNumber objects in the same field are automatically combined by arithmetic operations.
• The generator θ can be any algebraic number, represented in terms of radicals or Root objects. The coefficients ci must be integers or rational numbers.
• AlgebraicNumber is automatically reduced so that θ is an algebraic integer, and the list of ci is of length equal to the degree of the minimal polynomial of θ.
• AlgebraicNumber objects are always treated as numeric quantities.
• N finds the approximate numerical value of an AlgebraicNumber object.
• Operations such as Abs, Re, Round, and Less can be used on AlgebraicNumber objects.
• RootReduce can be used to transform AlgebraicNumber objects into Root objects.
• A particular algebraic number can have many different representations as an AlgebraicNumber object. Each representation is characterized by the generator θ specified for the field.
• AlgebraicNumber objects representing integers or rational numbers are automatically reduced to explicit integer or rational form.

# Examples

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## Basic Examples(1)

Represent an algebraic number:

Do arithmetic:

Get a numerical approximation:

## Scope(7)

AlgebraicNumber objects can be evaluated to any precision:

Objects representing integers or rational numbers are automatically simplified:

The generator θ in AlgebraicNumber[θ,{c0,,cn}] will automatically reduce to an algebraic integer:

Root objects:

AlgebraicNumber objects:

Coefficients of AlgebraicNumber objects are integers or rational numbers:

The number of coefficients is adjusted to match the degree of the algebraic number:

Arithmetic in a number field:

Operations on AlgebraicNumber objects:

## Applications(2)

Computations with AlgebraicNumber objects in the same number field are fast:

Make them part of the same number field:

In this example RootReduce automatically uses AlgebraicNumber object computations:

Compare to direct computations with Root objects:

Two solutions of the Pell equation :

More solutions can be deduced easily:

Check:

## Properties & Relations(5)

Use RootReduce to transform an algebraic number to a Root object:

Use ToNumberField to get representations of Root objects as AlgebraicNumber objects:

Get the generator polynomial:

Algebraic number theory operations:

Minimal polynomial:

## Possible Issues(1)

Operations such as Sqrt, Re, and Im do not automatically reduce:

Convert to AlgebraicNumber using RootReduce:

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

#### Text

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

#### CMS

Wolfram Language. 2007. "AlgebraicNumber." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AlgebraicNumber.html.

#### APA

Wolfram Language. (2007). AlgebraicNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AlgebraicNumber.html

#### BibTeX

@misc{reference.wolfram_2024_algebraicnumber, author="Wolfram Research", title="{AlgebraicNumber}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/AlgebraicNumber.html}", note=[Accessed: 18-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_algebraicnumber, organization={Wolfram Research}, title={AlgebraicNumber}, year={2007}, url={https://reference.wolfram.com/language/ref/AlgebraicNumber.html}, note=[Accessed: 18-May-2024 ]}