Ну это техническая задача.
Вот
здесь описан ручной метод. А вот как ее решает GAP с пакетом
RadiRoot:
Код:
GAP4, Version: 4.4.9 of 6-Nov-2006, x86_64-pc-linux-gnu-x86_64-linux-gnu-gcc
Components: trans 1.0 loaded.
Packages: Alnuth 2.2.5, Polycyclic 2.2 loaded.
gap> LoadPackage("radiroot");
-------------------------------------------------------------------------------------------------------------------------
Loading RadiRoot 2.2 (Roots of a Polynomial as Radicals)
by Andreas Distler (a.distler@tu-bs.de).
-------------------------------------------------------------------------------------------------------------------------
true
gap> x := Indeterminate( Rationals, "x" );;
gap> f := UnivariatePolynomial( Rationals, [-18,20,0,10,0,1] );
x^5+10*x^3+20*x-18
gap> IsSolvablePolynomial(f);
true
gap> RootsOfPolynomialAsRadicals( f, "latex" );
"/tmp/tmp.hDz4FG/Nst.tex"
Содержимое Nst.tex такое:
An expression by radicals for the roots of the polynomial
with the
-th root of unity
and
is: