Graduate Algebra (Hardcover)

Introduction to modules and their structure theory Introduction and prerequisites Exercises-Chapter 0 Part I. Modules: Finitely generated modules Simple modules and composition series Exercises-Part I Part II. Affine algebras and Noetherian rings: Galois theory of fields Algebras and affine fields Transcendence degree and the Krull dimension of a ring Modules and rings satisfying chain conditions Localization and the prime spectrum The Krull dimension theory of commutative Noetherian rings Exercises-Part II Part III. Applications to geometry and number theory: The algebraic foundations of geometry Applications to algebraic geometry over the rationals -- Diophantine equations and elliptic curves Absolute values and valuation rings Exercises-Part III List of major results Bibliography Index.