"A Classical Introduction to Modern Number Theory" is a revised and expanded version of the 1972 book "Elements of Number Theory" by Bogden and Quigley, Inc. Publishers. The authors, Kenneth Ireland and Michael Rosen, aim to provide a comprehensive introduction to modern number theory, focusing on algebraic number theory and arithmetic algebraic geometry. The book is designed for upper-level undergraduate mathematics majors and graduate students with a background in abstract algebra. The content covers a wide range of topics, including prime numbers, unique factorization, arithmetic functions, congruences, and the law of quadratic reciprocity. Later chapters delve into reciprocity laws, diophantine equations, and zeta functions. The book emphasizes the historical development of these topics and includes extensive references and exercises to enhance understanding and further study. Key themes include: 1. **Reciprocity Laws**: From quadratic reciprocity to the Artin reciprocity law. 2. **Diophantine Equations**: Over finite fields and rational numbers, covering topics like sums of squares and Fermat's Last Theorem. 3. **Zeta Functions**: Discussing the Riemann zeta function, Dirichlet L-functions, and zeta functions associated with algebraic curves. The book also features detailed historical notes and a comprehensive bibliography, making it a valuable resource for both students and researchers in number theory."A Classical Introduction to Modern Number Theory" is a revised and expanded version of the 1972 book "Elements of Number Theory" by Bogden and Quigley, Inc. Publishers. The authors, Kenneth Ireland and Michael Rosen, aim to provide a comprehensive introduction to modern number theory, focusing on algebraic number theory and arithmetic algebraic geometry. The book is designed for upper-level undergraduate mathematics majors and graduate students with a background in abstract algebra. The content covers a wide range of topics, including prime numbers, unique factorization, arithmetic functions, congruences, and the law of quadratic reciprocity. Later chapters delve into reciprocity laws, diophantine equations, and zeta functions. The book emphasizes the historical development of these topics and includes extensive references and exercises to enhance understanding and further study. Key themes include: 1. **Reciprocity Laws**: From quadratic reciprocity to the Artin reciprocity law. 2. **Diophantine Equations**: Over finite fields and rational numbers, covering topics like sums of squares and Fermat's Last Theorem. 3. **Zeta Functions**: Discussing the Riemann zeta function, Dirichlet L-functions, and zeta functions associated with algebraic curves. The book also features detailed historical notes and a comprehensive bibliography, making it a valuable resource for both students and researchers in number theory.