Keune F. Number Fields 2023

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Keune F. Number Fields 2023

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Textbook in PDF and DJVU formats This book is a textbook for algebraic number theory. It grew out of lecture notes of master courses taught at the Radboud University (Nijmegen) over a period of more than four decades. It is self-contained in the sense that it uses only mathematics of a bachelor level, including some Galois theory, as for example treated in Galois Theory [35]. To some extent the language of categories is used, especially in later chapters. Part I contains topics in basic algebraic number theory as they may be presented in a beginning master course on algebraic number theory. The theory in Part II is more advanced. It contains in particular full proofs of the main theorems of class field theory using a ‘classical’ approach to class field theory, which is in a sense a natural continuation of the basic theory in Part I. The advantage for students is that no more prerequisites are needed. Each approach has its own advantages, so for specialists in algebraic number theory it is advisable to have knowledge of more than just one. For specialists of other areas of pure mathematics who want to use it, the exposition as given here might very well suffice. The last two chapters provide the connection to the more modern and more advanced id`elic version of class field theory. It is not the purpose of this book to present up to date information on the state of the art. The section References is just what it says: it contains references made in the text and is not an exposition of the vast literature on the subject