%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %W radiroot.tex Radiroot documentation Andreas Distler %% %H $Id: radiroot.tex,v 1.6 2008/01/22 11:57:43 gap Exp $ %% %Y 2005 %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Functionality of the Package} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This chapter describes the methods available in the {\Radiroot} package. \Section{Methods for Rational Polynomials} \> IsSeparablePolynomial( <f> ) returns `true' if the rational polynomial <f> has simple roots only and `false' otherwise. \> 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. \> SplittingField( <f> ) \> IsomorphicMatrixField( <F> ) \> RootsAsMatrices( <f> ) \> IsomorphismMatrixField( <F> ) For a normed, rational polynomial <f>, `SplittingField(<f>)' returns the smallest algebraic extension field <L> of the rationals containing all roots of <f>. The field is constructed with `FieldByPolynomial' (see Creation of number fields in \Alnuth). The primitive element of <L> is denoted by `a'. A matrix field <K> isomorphic to <L> is known after the computation and can be accessed using `IsomorphicMatrixField(<L>'. The matrices, one for each distinct root of <f>, in the list produced by `RootsOfMatrices(<f>)' lie in <K>. `IsomorphismMatrixField( <L> )' returns an isomorphism of <L> onto <K>. \beginexample gap> x := Indeterminate( Rationals, "x" );; gap> f := UnivariatePolynomial( Rationals, [1,3,4,1] ); x^3+4*x^2+3*x+1 gap> L := SplittingField( f ); <algebraic extension over the Rationals of degree 6> gap> y := Indeterminate( L, "y" );; gap> FactorsPolynomialAlgExt( L, f ); [ y+((-168/47-535/94*a-253/94*a^2-24/47*a^3-3/94*a^4)), y+((20/47-441/94*a-253/94*a^2-24/47*a^3-3/94*a^4)), y+((336/47+488/47*a+253/47*a^2+48/47*a^3+3/47*a^4)) ] gap> IsomorphicMatrixField( L ); <rational matrix field of degree 6> gap> Display(RootsAsMatrices(f)[1]); [ [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ -1, -3, -4, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, -1, -3, -4 ] ] gap> MinimalPolynomial( Rationals, RootsAsMatrices(f)[1]); x^3+4*x^2+3*x+1 gap> iso := IsomorphismMatrixField( L ); MappingByFunction( <algebraic extension over the Rationals of degree 6>, <rational matrix field of degree 6>, function( x ) ... end, function( mat ) ... end ) gap> PreImages( iso, RootsAsMatrices( f ) ); [ (-336/47-488/47*a-253/47*a^2-48/47*a^3-3/47*a^4), (-20/47+441/94*a+253/94*a^2+24/47*a^3+3/94*a^4), (168/47+535/94*a+253/94*a^2+24/47*a^3+3/94*a^4) ] \endexample To factorise a polynomial over its splitting field one has to use `FactorsPolynomialAlgExt' (see \Alnuth) instead of `Factors'. \> GaloisGroupOnRoots( <f> ) calculates the Galois group <G> of the rational polynomial <f>, which has to be separable, as a permutation group with respect to the ordering of the roots of <f> given as matrices by `RootsAsMatrices'. \beginexample gap> GaloisGroupOnRoots(f); Group([ (2,3), (1,2) ]) \endexample If you only want to get the Galois group abstractly, and if $f$ is irreducible of degree at most 15, it is often better to use the function `GaloisType' (see Chapter~"ref:Polynomials over the Rationals" in the {\GAP} reference manual). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Solving a Polynomial by Radicals} \> RootsOfPolynomialAsRadicals( <f> [, <mode> [, <file> ] ] ) computes a solution by radicals for the irreducible, rational polynomial <f> up to degree 15 if the Galois group of <f> is solvable, and returns `fail' otherwise. If it succeeds and <mode> is not `off', the function returns the path to a file containing the description of the roots of <f> and generators of cyclic radical extensions to produce its splitting field. The user has several options to specify what happens with the results of the computation. Therefore the optional second argument <mode>, a string, can be set to one of the following values: \beginexample "dvi" \endexample Provided `latex' and the dvi viewer `xdvi' are available, this option will display the irreducible radical expression for the roots and cyclic extension generators in a new window. The package uses this option as the default. \beginexample "latex" \endexample A LaTeX file is generated which contains the encoding for the expression by radicals. This gives the user the opportunity to adjust the layout of the individual example before displaying the expression. \beginexample "maple" \endexample The generated file can be read into Maple \cite{Maple10} which makes a root of <f> available as variable `a'. \beginexample "off" \endexample In this mode the function does not actually compute a radical expression but is only called for its side effects. Namely, the attributes `SplittingField', `RootsAsMatrices' and `GaloisGroupOnRoots' are known for <f> afterwards. This is slightly more effective than calling the corresponding operations one by one. With the optional third argument <file> the user can specify a file name under which the description files will be stored in the directory from which \GAP\ was called. Depending on the option for <mode> an extension like `.tex' might be added automatically. If <file> is not given, the function places description files in a new directory `/tmp/tmp.'<string> with names such as `Nst' and `Nst.tex'; the temporary directory is removed at the end of the {\sf GAP} session. %Note that it is not possible to have <file> as the second argument. The computation may take a very long time and can get unfeasible if the degree of <f> is greater than 7. \beginexample \endexample \> RootsOfPolynomialAsRadicalsNC( <f> [, <mode> [, <file> ] ] ) does essentially the same as `RootsOfPolynomialAsRadicals' except that it runs no test on the input before starting the actual computation. Therefore it can be used for polynomials with arbitrary degree, but it may run for a very long time until a non-solvable polynomial is recognized as such. Detailed examples for these two functions can be found in the next section. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Examples} The function `RootsOfPolynomialAsRadicals' does not generate output inside \GAP. Depending on the chosen mode, various kinds of files can be created. As an example the polynomial from the introduction will be considered. \beginexample gap> g := UnivariatePolynomial( Rationals, [1,1,-1,-1,1] ); x^4-x^3-x^2+x+1 gap> RootsOfPolynomialAsRadicals(g); "/tmp/tmp.8zkw5B/Nst.tex" \endexample will cause a dvi file to appear in a new window: An expression by radicals for the roots of the polynomial $x^{4}-x^{3}-x^{2} + x + 1$ with the $n$-th root of unity $\zeta_n$ and $\omega_1 = \sqrt{ - 3}$, $\omega_2 = \sqrt{\frac{7}{2} - \frac{1}{2}\omega_1}$, $\omega_3 = \sqrt{\frac{7}{2} + \frac{1}{2}\omega_1}$, is: $\frac{1}{4} - \frac{1}{4}\omega_1 + \frac{1}{2}\omega_2$ If one wants to work with the roots, it might be helpful to use Maple \cite{Maple10}, in which an expression like $2^{(1/2)}$ is valid. \beginexample gap> RootsOfPolynomialAsRadicals(g, "maple"); "/tmp/tmp.k9aTCz/Nst" \endexample will create a file with the following content: \beginexample w1 := (-3)^(1/2); w2 := ((7/2) + (-1/2)*w1)^(1/2); w3 := ((7/2) + (1/2)*w1)^(1/2); a := (1/4) + (1/4)*w1 + (1/2)*w3; \endexample After those computations several attributes are known for the polynomial in \GAP. \beginexample gap> RootsOfPolynomialAsRadicalsNC( g, "off" ); gap> time; 0 gap> SplittingField( g ); <algebraic extension over the Rationals of degree 8> gap> time; 0 gap> GaloisGroupOnRoots( g ); Group([ (2,4), (1,2)(3,4) ]) gap> time; 0 \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E