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2 Functionality of the package

Sections

  1. Methods for rational polynomials
  2. Solving a polynomial by radicals

2.1 Methods for rational polynomials

  • IsSolvable( f )
  • IsSolvablePolynomial( f )

    returns true if the rational polynomial f has a solvable Galois group and false otherwise. It signals an error if there exists an irreducible factor with degree greater than 15.

    For a rational polynomial f

  • SplittingField( f )

    returns the smallest field, constructed with FieldByPolynomial, that contains all roots of f.

  • GaloisGroupOnRoots( f )

    calculates the Galois group G of the rationals polynomial f as a permutation group with respect to the ordering of the roots of f given as matrices in G!.roots.

    2.2 Solving a polynomial by radicals

  • RootsOfPolynomialAsRadicals( f )

    computes a solution by radicals for the irreducible, rational polynomial f up to degree 15 if this is possible, e. g. if the Galoisgroup of f is solvable. The result is displayed in form of a dvi-file. Additionally a record is returned which contains the roots of f as a list roots of matrices, the Galois group on the roots as component galgrp and the splitting field of f in two forms; on the one hand the matrix field K generated by the roots and on the other hand an algebraic number field H created by the defining polynomial of the matrix field. The record also includes a list cyclics of matrices which define the splitting field by gradual, cyclic extensions.

    The computation may last very long and doesn't finish for every example if the degree of f is greater than 7.

  • RootsOfPolynomialAsRadicalsNC( f, display )

    does essentially the same as RootsOfPolynomialAsRadicals except that you can choose if you want to create a dvi-file and display it by setting the boolean display. The function performs no test whether the polynomial f is irreducible. It also doesn't check at the begining if f is solvable, but can therefore be used for polynomials with arbitrary degree.

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    Radiroot manual
    July 2005