Number theory

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Algebraic number theory is a sophisticated area of mathematics shaped by prominent mathematicians throughout history. This book aims to present the core elements of algebraic number theory, including normal extensions and an introduction to class field theory. It treats algebraic functions alongside algebraic numbers, highlighting the analogy between number fields and function fields, particularly when the ground field is finite. This approach also introduces 'higher congruences,' vital for 'arithmetic geometry.' Early chapters cover elementary number theory topics like Minkowski's geometry of numbers, public-key cryptography, and a concise proof of the Prime Number Theorem, inspired by Newman and Zagier. Subsequent chapters introduce essential tools such as ideals, discriminants, and valuations, applying these to function fields, including a proof of the Riemann-Roch Theorem and the first case of Fermat's Last Theorem via cyclotomic fields. The book also details Hecke $L$-series theory, following Tate, with applications to number theory, including the Generalized Riemann Hypothesis. Chapter 9 consolidates earlier material through quadratic number fields, while Chapter 10 offers an introduction to class field theory. The text aims to provide straightforward proofs, making it accessible for beginners eager to explore the depth of the subject. It is structured for two one-semester courses, with the first four chapters establishi