Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers rational numbers and their generalizationsnumber theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers finite fields and function fieldsthese properties such as whether a ring admits . The techniques of elementary arithmetic ring theory and linear algebra are shown working together to prove important theorems such as the unique factorization of ideals and the finiteness of the ideal class group the book concludes with two topics particular to quadratic fields continued fractions and quadratic forms. Of course the idea of studying quadratic extensions as a way of learning the basics of algebraic number theory is not new to this author and he certainly does not claim that it is the classic text introduction to the theory of numbers by niven zuckerman and montgomery has been around for decades now and contains a chapter on algebraic . Chapter 1 number fields 5 1 example quadratic number elds 5 2 complex embeddings 8 3 example cyclotomic elds 9 4 galois theory of number elds 14 5 relative extensions 17 6 exercises 22 chapter 2 rings of integers 25 1 unique factorization 25 2 algebraic integers 33 3 unique factorization of ideals in dedekind domains 43 4
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