Gupta, Vaibhav

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.