The Algebraic package is a package for computing with polynomials and rational functions whose coefficients are algebraic numbers, in arbitrary characteristic. The main functionality is computing normal forms and greatest common divisors.

Algebraic numbers and functions can be represented by radicals or by RootOfs, or a mixture of them. Nested radicals or RootOfs are supported as well.

Non-algebraic expressions, such as, e.g., powers with non-rational exponents or special functions, are not directly supported by the routines in this package. Such subexpressions will be frozen and, where appropriate, their arguments handled recursively and independently.

To avoid ambiguities, it is recommended that all RootOfs are indexed.

The following is a list of available commands.

Content - extract the greatest common divisor of the coefficients of a polynomial

ConvertRootOf - convert algebraic objects to RootOf representation

Degree - compute the formal algebraic degree of a tower of extensions

Divide - test for divisibility of multivariate polynomials

Expand - expand products and powers into a sum

ExtendedEuclideanAlgorithm - perform the extended Euclidean algorithm on a pair of polynomials

GetAlgebraics - extract all algebraic objects from a given object

GreatestCommonDivisor - compute the greatest common divisor and cofactors of two polynomials

MakeMonic - rewrite a RootOf as an equivalent monic RootOf

Normal - determine the normal form of a rational function with algebraic coefficients

PrimitivePart - remove all common factors from the coefficients of a polynomial

PseudoDivision - perform fraction-free division on a pair of polynomials

Quotient - compute the quotient when Euclidean division is performed on two polynomials

Reduce - reduce powers of RootOfs modulo their defining polynomials

Remainder - compute the remainder when Euclidean division is performed on two polynomials

Resultant - determine the resultant of two polynomials with respect to a variable

Squarefree - factor a polynomial into a product of powers of square-free polynomials.

To display the help page for a particular Algebraic command, see Getting Help with a Command in a Package.