Thanks to2025年10月02日更新
[1]In January of this year (2025), I started a project that I had wanted
to try for a long time, "using Mathematica to find radical solutions
of higher-order equations (fifth-order or higher). The first project I
started was a quintic equation (5th order equation), and it took me about
a month to complete the program, which was based on the Galois group and
primitive element , which was quite time consuming to solve. Then in April
I solved the seventh order equation in the same way (but using SageMath
for the Galois group & primitive elements). In May I started working
on the sextic equation (sixth order equation). However, SageMath could
not find the primitive elements of the sextcs, so I learned how to use
Galois Resolvent from a paper by Prof. Hagedoen and was able to solve the
sextics eventuallly. In June, I tried to solve quintic equation using Galois
Resolvent, and finally on July 7, I was able to solve quintic equation
using Galois Resolvent. The method using Galois Resolvent takes more time
to derive the auxiliary equations than the method using the primitive element&Galois
Group. But once the auxiliary equations are obtained, the actual solution
is very fast. In fact, when I solved the quintic equation using primitive
elements, it took me 30 seconds per problem, but using Galois Resolvent,
I could solve it in 0.5 seconds. (^^♪
[2] Next, I tried to solve a septic (seventh order equation) using Galois
Resolvent, but it did not work because the order of the auxiliary equation
in the case of a septic is 120. So, for a change, I decided to publish
an English version of the programs. Since I have created about 20 files
and it would be too much work to translate them all, I decided to upload
one app for each of the quitic and sextic equations, which can be run with
the free Wolfram Player. Wolfram Player is available for Windows/Mac/Linux/iOS, so you should
be able to run it on your iPhone or iPAD. (I do not recommend using an
iPhone, however, because the exact solution is longer.) A Windows example
can be found here(quintic) , here(sextic).
[3]Both apps allow you to download the file, open it with Wolfram Player,
and type in the coefficients of f(x) to get the exact solution in an instant.
[4] If you are interested in my application, I have detailed files in Japanese here(quintic) and here(sextic). DeepL or Google Translatator will give you some idea.
[5]Please use the email link below for any comments or questions. Enjoy!
I hope this will encourage more people to become enthusiasts of Mathematica,
GeoGebra, Magma, SageMath, and other mathematical software.
P.S.[6] [2nd Ocb. 2025] I have written a program to find a radical solution
to a solvable septic equation, following the method described in Ref. [4].
A Windows example can be found here (septic). It is an elegant and relatively concise solution. The same authors also introduced a new method for finding a radical solution to a quintic equation. Ref. [4] also uses lots of wonderful method -such as Groebner basis, resultant, new simple resolvents. It is definitely worth reading, although it contains many small errors. (For example, in equation (19), "-1/7c5" should be "1/7c5"; and in equation (22), "-1/49b42" should be "1/49b42".)
Thanks to
App (needs free Wolfram Player)
| Notebook | |
| Quintic Equation | solveSexticProgram.nb |
| Sextic Equation | solveSexticApp.nb |
| Septic Equation | solveSeptcApp.nb |
List of solvable irreducible quintics | solvableQuintic.txt |
List of solvable irreducible sextics| solvableSextic.txt |
List of solvable irreducible sextics Top page solving 5,6,7th degree polynomial equations by Mathematica  email me |