Master of Science (MS)
Abstract Algebra, Galois Theory, Groups, Hilbert, Irreducibility, Theorem
We study abstract algebra and Hilbert's Irreducibility Theorem. We give an exposition of Galois theory and Hilbert's Irreducibility Theorem: given any irreducible polynomial f(t1, t2, …, tn, x) over the rational numbers, there are an infinite number of rational n-tuples (a1, a2, …, an) such that f(a1, a2, …, an, x) is irreducible over the rational numbers.
We take a preliminary look at linear algebra, symmetric groups, extension fields, splitting fields, and the Chinese Remainder Theorem. We follow this by studying normal extension fields and Galois theory, proving the fundamental theorem and some immediate consequences. We expand on Galois theory by
exploring subnormal series of subgroups and define solvability with group property P, ultimately proving Galois' Theorem. Beyond this, we study symmetric functions and large extension fields with Galois group Sn.
We detour into complex analysis, proving a few of Cauchy's theorems, the identity theorem, which is a key to proving Hilbert's Irreducibility Theorem, and meromorphic functions. We study affine plane curves, regular values, and the Density Lemma — which bounds the rational outputs a non-rational meromorphic function has for rational inputs. Ultimately, we prove the Hilbert Irreducibility Theorem and apply it to symmetric functions to construct fields whose Galois group is Sn.
Adams, Damien, "Galois Theory and the Hilbert Irreducibility Theorem" (2013). Master's Theses. 4256.