2025年07月11日更新
Solving quintics and sextics in radicals with Mathematica
[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) and 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.
Thanks to
App (needs free Wolfram Player)
List of solvable irreducible quintics
List of solvable irreducible sextics
By Shigeki Ogose All rights reserved
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