Algebraic Number Objects
Mathematica 5 brings high-performance algebraic number
arithmetic within specified algebraic
number fields. Mathematica has been providing
representation of algebraic numbers as Root
objects since Version 3. A Root object contains the minimal
polynomial of the algebraic number and the root number--an integer
indicating which of the roots of the minimal polynomial
the Root object represents. This allows for unique
representation of arbitrary complex algebraic numbers.
A disadvantage is that performing arithmetic operations in this
representation is costly. If we restrict ourselves to computations
within a fixed finite algebraic extension,  , of rationals, we can use a
more convenient representation of elements of , namely as polynomials in
 . Within a fixed algebraic
number field, the algebraic number arithmetic in the Algebraic object representation is much
faster than in the Root object
representation. Mathematica 5 contains extensive functionality
related to algebraic number fields, available through the package NumberTheory`AlgebraicNumberFields`.
Suppose we have the following algebraic numbers:
![{x, y, z} = {i, 2^(1/2), Root[#1^3 - 2 #1 + 3 &, 1]} ;](images/numberobjects_2.gif "{x, y, z} = {i, 2^(1/2), Root[#1^3 - 2 #1 + 3 &, 1]} ;")
And want to compute values of  .
A direct computation of the value using RootReduce takes a rather long time.
![Timing[RootReduce[(-2 y z (7 + x - y + z^2) + (6 + x^2 + 2 y) (-11 + x y + z^2))/(2 y z (-4 - x + 3 y z) - (6 + x^2 + 2 y) (2 - 2 x + z^3))] ;]](images/numberobjects_4.gif "Timing[RootReduce[(-2 y z (7 + x - y + z^2) + (6 + x^2 + 2 y) (-11 + x y + z^2))/(2 y z (-4 - x + 3 y z) - (6 + x^2 + 2 y) (2 - 2 x + z^3))] ;]")
A faster alternative is to first find a common algebraic number field containing
{a, b, c}, which takes only a fraction of a
second.
![Timing[({x, y, z} = ToCommonField[{x, y, z}]) ;]](images/numberobjects_6.gif "Timing[({x, y, z} = ToCommonField[{x, y, z}]) ;]")
Arithmetic within the common number field is much faster.
![Timing[(-2 y z (7 + x - y + z^2) + (6 + x^2 + 2 y) (-11 + x y + z^2))/(2 y z (-4 - x + 3 y z) - (6 + x^2 + 2 y) (2 - 2 x + z^3)) ;]](images/numberobjects_8.gif "Timing[(-2 y z (7 + x - y + z^2) + (6 + x^2 + 2 y) (-11 + x y + z^2))/(2 y z (-4 - x + 3 y z) - (6 + x^2 + 2 y) (2 - 2 x + z^3)) ;]")
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