This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# AlgebraicNumber

 AlgebraicNumberrepresents the algebraic number in the field given by .
• AlgebraicNumber objects in the same field are automatically combined by arithmetic operations.
• The generator can be any algebraic number, represented in terms of radicals or Root objects. The coefficients must be integers or rational numbers.
• AlgebraicNumber is automatically reduced so that is an algebraic integer, and the list of is of length equal to the degree of the minimal polynomial of .
• A particular algebraic number can have many different representations as an AlgebraicNumber object. Each representation is characterized by the generator specified for the field.
• AlgebraicNumber objects representing integers or rational numbers are automatically reduced to explicit integer or rational form.
Represent an algebraic number:
Do arithmetic:
Get a numerical approximation:
Represent an algebraic number:
 Out[1]=
Do arithmetic:
 Out[2]=
Get a numerical approximation:
 Out[3]=
 Scope   (7)
AlgebraicNumber objects can be evaluated to any precision:
Objects representing integers or rational numbers are automatically simplified:
The generator in AlgebraicNumber will automatically reduce to an algebraic integer:
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 :
More solutions can be deduced easily:
Check:
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:
Operations such as Sqrt, Re, and Im do not automatically reduce:
Convert to AlgebraicNumber using RootReduce:
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