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AlgebraicNumber

AlgebraicNumber[Theta, {c0, c1, ..., cn}]
represents the algebraic number in the field Q[theta] given by c_0+c_1theta +...+c_n theta^n.
  • AlgebraicNumber objects in the same field are automatically combined by arithmetic operations.
  • The generator Theta 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 Theta is an algebraic integer, and the list of ci is of length equal to the degree of the minimal polynomial of Theta.
  • A particular algebraic number can have many different representations as an AlgebraicNumber object. Each representation is characterized by the generator Theta specified for the field.
  • AlgebraicNumber objects representing integers or rational numbers are automatically reduced to explicit integer or rational form.
EXAMPLESCLOSE ALL
Basic Examples   (1)
Represent an algebraic number:
Do arithmetic:
Get a numerical approximation:
Represent an algebraic number:
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Do arithmetic:
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Get a numerical approximation:
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Scope   (7)
AlgebraicNumber objects can be evaluated to any precision:
Objects representing integers or rational numbers are automatically simplified:
The generator Theta in AlgebraicNumber[Theta, {c0, ..., cn}] will automatically reduce to an algebraic integer:
Radical expressions:
Root 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 x^2-82 y^2=+/-1:
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:
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