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:

Radical expressions:

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:

Introduced in 2007
 (6.0)