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Algebraic number theory is the branch of number theory that deals with algebraic numbers. Historically, algebraic number theory developed as a set of tools for solving problems in elementary number theory, namely Diophantine equations (i.e., equations whose solutions are integers or rational numbers). More recently, algebraic number theory has developed into the abstract study of algebraic numbers and number fields themselves, as well as their properties. Algebraic number theory is a major branch of number theory that studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field extensions. In this setting, the familiar features of the integers—such as unique factorization—need not hold. The virtue of the primary machinery employed—Galois theory, group cohomology, group representations, and L-functions—is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.

The higher reaches of algebraic number theory are now one of the crown jewels of mathematics. But algebraic number theory is not merely interesting in itself. It has become an important tool over a wide range of pure mathematics, and many of ideas involved generalize, for example to algebraic geometry. This book is intended both for number theorist and more generally for working algebraists.