This section is provided by Raoof Mirzaei
City of Knowledge Ketab.io.
Raoof Mirzaei is a researcher in mathematics and mathematical physics. He was born in 1995 in Kermanshah, Iran. His interests are in Group theory, Galois theory, and Algebraic and Differential geometry.
The aim of this series (Foundations of Abstract Mathematics) is to try and start from a very basic level so that even undergraduate students can follow the subject. Along the way we will talk about group theory and Galois theory and try to show that group theory can be modeled very differently from what Galois did in showing the unsolvability of the quintic equation, and then we will see that other consequences occur.
For these purposes we need to have some information from Model theory and Category theory and use dualities to show that the Galois idea which uses the relations between groups and fields is not what it seems to be to make the quintic unsolvable.
In these parts I will use Superexponential Algebra results and make connections with them.
We mention Hermite's method of using elliptic functions and some other ways for solving quintic equations. Then we will attack quintics using group theory.
After that we will try to make more of a connection between geometry and group theory using the concept of symmetries which can be viewed as a version of the Erlangen program.
1. Foundations of Abstract Mathematics.
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