Galois Theory and Hilbert Irreducibility Theorem
Extended draft version of the master thesis
Publication date
2025
Document type
Master thesis
Thesis type
Masterarbeit
Author
Advisor
Jana, Ranjan Kumar
Thangadurai, Ravindranathan
Granting institution
Sardar Vallabhbhai National Institute of Technology
Exam date
2023-05-18
Organisational unit
Sardar Vallabhbhai National Institute of Technology, Surat, India
Publisher
Sardar Vallabhbhai National Institute of Technology
Part of the university bibliography
Nein
File(s)
Language
English
Keyword
Galois Theory
Hilbert Irreducibility Theorem
Abstract
In this dissertation, we are studying abstract algebra; mostly, our focus is on studying Galois theory in depth, and then we will study the proof of our main theorem, "Hilbert Irreducibility Theorem," which states that given any irreducible polynomial g (t_1, t_2,...., t_n, x) over the rational numbers, there are an infinite number of rational n-tuples (a_1, a_2,...., a_n) such that f (a_1, a_2,..., a_n, x) is irreducible over the rational numbers. I have omitted the basics of abstract algebra, such as group, ring, and field theory, and motivated the reader to read a basic book to learn these topics. I have presumed that the reader knows linear algebra. I started my dissertation with an introduction to group characters and then extended our discussion to Galois extension and normal extension to provide the basis for studying the Fundamental Theorem of Galois Theory. Then, we fixed our focus on Kummer Extensions and Cyclotomic Extensions. To end our discussion of Galois' theory, we studied solvable groups. Then some complex analysis theorems have been stated, which we will use in our proof of Hilbert's irreducibility theorem. A whole chapter has been dedicated to studying lemmas to prove our theorem, and then in the last chapter, we have proved our theorem.
Description
This is the extended draft version of the author's master's thesis titled "A Study of Hilbert Irreducibility Theorem" written at Sardar Vallabhbhai National Institute of Technology, Surat, and Harish Chandra Research Institute, Prayagraj, India, in 2023.
Version
Draft
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