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Radiroot manual
The info mechanism in GAP allows functions to print information
during the computation (see Section Info Functions in the
GAP reference manual for general information).
InfoRadiroot V
is the info class of this package.
SetInfoLevel( InfoRadiroot, level )
sets the info level for InfoRadiroot to level, where level has
to be an integer in the range 0-4.
The default value for InfoRadiroot is 1. Information why a function
returns fail will be given with this setting.
gap> InfoLevel(InfoRadiroot); 1 gap> RootsOfPolynomialAsRadicals(x^5-4*x+2); #I Polynomial is not solvable. fail
Setting the info level to a higher value will cause messages to show up during single steps of the computation. On level 2 one gets a rough overview. Those who want to go into the details of the algorithm described in Distler05 and of the implementation itself will find the information on level 3-4 helpful.
To use the package in silent mode the info level can be given the value 0.
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Radiroot manual
This package is available at
https://gap-packages.github.io/radiroot/
in form of a gzipped tar-archive. For installation instructions see
Chapter Installing a GAP Package in the GAP reference manual.
Normally you will unpack the archive in the pkg directory of your
GAP version by typing:
bash> tar xfz radiroot-2.9.tar.gz # for the gzipped tar-archive
To use the Radiroot package you have to request it explicitly. This is done by calling
gap> LoadPackage("radiroot");
-----------------------------------------------------------------------------
Loading RadiRoot 2.9 (Roots of a Polynomial as Radicals)
by Andreas Distler (a.distler@tu-bs.de).
Homepage: https://gap-packages.github.io/radiroot/
-----------------------------------------------------------------------------
true
The LoadPackage command is described in Section LoadPackage in
the GAP reference manual.
If you want to load the Radiroot package by default, you can put the
LoadPackage command into your gaprc file (see Section The gaprc file in the GAP reference manual).
Once the package is loaded, it is possible to check the correct installation by running the test suite of the package with the command
gap> ReadPackage( "radiroot", "tst/testall.g" );
To use Radiroot the package Alnuth in version 3.0 or higher has to be loaded with its interface fully functional.
In the standard mode a dvi file is created to display the roots of a
polynomial. As default the package uses the command latex searched
for in your system programs to create the dvi file and the command
xdvi to start the dvi viewer. If you can not use this settings you
will have to change the function RR_Display in the file Strings.gi
in the subdirectory lib of the package.
Radiroot manual
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Radiroot manual
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.
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+(-3/94*a^4-24/47*a^3-253/94*a^2-535/94*a-168/47), y+(-3/94*a^4-24/47*a^3-253/94*a^2-441/94*a+20/47), y+(3/47*a^4+48/47*a^3+253/47*a^2+488/47*a+336/47) ] 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 ) ); [ -3/47*a^4-48/47*a^3-253/47*a^2-488/47*a-336/47, 3/94*a^4+24/47*a^3+253/94*a^2+441/94*a-20/47, 3/94*a^4+24/47*a^3+253/94*a^2+535/94*a+168/47 ]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.
gap> GaloisGroupOnRoots(f); Group([ (2,3), (1,2) ])
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 Polynomials over the Rationals in the GAP reference manual).
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:
"dvi"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.
"latex"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.
"maple"The generated file can be read into Maple Maple10 which makes a root of f available as variable
a.
"off"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 GAP
session.
The computation may take a very long time and can get unfeasible if the degree of f is greater than 7.
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.
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Radiroot manual
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This package provides functionality to deal with one of the fundamental
problems in algebra. The roots of a rational polynomial shall be
expressed by radicals. This means one is only allowed to use the four
basic operations (+, −, · ,÷) and to extract roots. For
example, a radical expression for the roots of the polynomial x4 − x3 − x2 + x + 1 is
|
There are formulas to solve the general equation xn+ an−1xn−1+ ... + a1x+a0 = 0 up to degree 4. For higher degrees such formulae do not exist (Abel26). It was Évariste Galois (1811 -- 1832) who discovered that there exists a radical expression for the roots if and only if the Galois group of the polynomial - initially a permutation group on the roots - is solvable Galois97. But the task itself was impractical in his days. This package is the first public tool which provides a practical method for solving a polynomial algebraically. The implementation is based on Galois' ideas and the algorithm is described in Distler05.
The package can provide the result in various forms. As a default an expression is given in a similar way as in the example above. Alternatively, a file containing the roots might be created which is readable by Maple Maple10. In GAP itself some information deduced during the computation is available.
The user should be aware that radical expressions can get very complicated even for polynomials of small degree. Especially because the algorithm will find an irreducible radical expression. That means one gets a root of the given polynomial for every choice of a value of the radicals in the expression. Moreover it is not the aim of this package to give a simplest expression, in any sense.
In Chapter 2 the methods provided by this package are listed and explained.
Chapter 3 gives details about the info class of this package. See Section Info Functions in the GAP reference manual for general information about info classes.
While the installation of the package follows standard GAP rules the Chapter 4 contains information about external programs required by Radiroot in its default setup.
This package uses the interface in the package Alnuth, to factorise polynomials over algebraic number fields. This functionality must be available to use the functions in Radiroot.
This package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
Radiroot manual
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Radiroot manual
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