Description: This book deals with classical Galois theory, of both finite and infinite extensions, and with transcendental extensions, focusing on finitely generated extensions and connections with algebraic geometry. The purpose of the book is twofold. First, it is written to be a textbook for a graduate-level course on Galois theory or field theory. Second, it is designed to be a reference for researchers who need to know field theory. The book is written at the level of students who have familiarity with the basic concepts of a group, ring and vector space theory (including the Sylow theorems), factorization in polynomial rings, and theorems about bases of vector spaces. Readers who do not have the proper background can consult the appendices on ring theory, set theory, group theory, and vector spaces; these appendices provide the background necessary to understand the book. This book features a large number of examples and exercises, covers a large number of topics, and in most cases provides complete proofs for the stated results. To help readers grasp field theory, many concepts are placed in the context of their relationships with other areas of mathematics.

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