Quadratic Number Fields
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Beschreibung
This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level.
Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study.
Assuming a moderate background in elementarynumber theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.
This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level.
Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study.
Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.
Connects quadratic fields with modern algebraic number theory Applies the theory to solve Diophantine equations Contains hundreds of exercises with solutions Includes original historical commentary
Autor*in
Franz Lemmermeyer
Themen in »Quadratic Number Fields«
quadratic fields Pell equation class group Gauss sum Pell conics modularity Fermat's last theorem Catalan's conjecture Fekete polynomials
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“This is a wonderful, well-written introduction to modern algebraic number theory that has been made accessible to a broad undergraduate audience through the author’s restriction to quadratic number fields. Historically motivated, it shows how algebraic number theory has evolved over time and depicts it as living and breathing, not as a field that became static in the late 19th century.” (Benjamin Linowitz, MAA Reviews, November 21, 2023)
“The book is very nicely written and the original style and choices of the topics make it agreeable reading, and might well complement and motivate the study of other classical introductions to the theory of more general number fields.” (Alessandro Cobbe, zbMATH 1498.11003, 2022)
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