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 )
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.
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