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Die Theorie der quadratischen Zahlkörper ist der erste Schritt hin auf eine allgemeine Theorie algebraischer Zahlkörper. In diesem Buch werden die Hauptsätze der Theorie nicht auf dem kürzesten Weg bewiesen; vielmehr nehmen wir uns die Zeit, uns auf kleinen Umwegen mit den neuen Objekten vertraut zu machen und die Sätze an vielen Beispielen zu illustrieren. Außerdem gehen wir ausführlich auf die Geschichte der algebraischen Zahlentheorie ein und besprechen einige für die Entwicklung dieser Disziplin wichtige Beispiele. Dabei spielen vor allem diophantische Gleichungen eine große Rolle. Abgerundet wird das Buch durch zahlreiche Übungsaufgaben und eine kurze Einführung in das Rechnen mit Pari und Sage. Der Autor Franz Lemmermeyer hat nach seiner Promotion in Heidelberg und seiner Habilitation in Bonn an Universitäten in den USA und in der Türkei gelehrt, und unterrichtet seit 2007 Mathematik am Gymnasium St. Gertrudis in Ellwangen.

Commutative algebra. --- Commutative rings. --- Number theory. --- Commutative Rings and Algebras. --- Number Theory.

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This volume presents a collection of articles highlighting recent developments in commutative algebra and related non-commutative generalizations. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non-Noetherian ring theory, module theory and integer-valued polynomials along with connections to algebraic number theory, algebraic geometry, topology and homological algebra. Most of the eighteen contributions are authored by attendees of the two conferences in commutative algebra that were held in the summer of 2016: “Recent Advances in Commutative Ring and Module Theory,” Bressanone, Italy; “Conference on Rings and Polynomials”  Graz, Austria. There is also a small collection of invited articles authored by experts in the area who could not attend either of the conferences. Following the model of the talks given at these conferences, the volume contains a number of comprehensive survey papers along with related research articles featuring recent results that have not yet been published elsewhere.

Mathematics. --- Associative rings. --- Rings (Algebra). --- Commutative algebra. --- Commutative rings. --- Commutative Rings and Algebras. --- Associative Rings and Algebras. --- Commutative rings --- Rings (Algebra) --- Algebra. --- Mathematics --- Mathematical analysis --- Algebraic rings --- Ring theory --- Algebraic fields --- Algebra

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This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group. Applications involving symm etry groups, determinants, linear coding theory and cryptography are interwoven throughout. Each section ends with ample practice problems assisting the reader to better understand the material. Some of the applications are illustrated in the chapter appendices. The author's unique melding of topics evolved from a two semester course that he taught at the University of British Columbia consisting of an undergraduate honors course on abstract linear algebra and a similar course on the theory of groups. The combined content from both makes this rare text ideal for a year-long course, covering more material than most linear algebra texts. It is also optimal for independent study and as a supplementary text for various professional applications. Advanced undergraduate or graduate students in mathematics, physics, computer science and engineering will find this book both useful and enjoyable.

Mathematics. --- Algebraic geometry. --- Commutative algebra. --- Commutative rings. --- Group theory. --- Matrix theory. --- Algebra. --- Commutative Rings and Algebras. --- Linear and Multilinear Algebras, Matrix Theory. --- Group Theory and Generalizations. --- Algebraic Geometry. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebraic geometry --- Math --- Geometry, algebraic. --- Mathematics --- Mathematical analysis --- Algebra --- Geometry --- Group theory --- Vector spaces --- Matrices --- Rings (Algebra)

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This book presents state-of-the-art research and survey articles that highlight work done within the Priority Program SPP 1489 “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory”, which was established and generously supported by the German Research Foundation (DFG) from 2010 to 2016. The goal of the program was to substantially advance algorithmic and experimental methods in the aforementioned disciplines, to combine the different methods where necessary, and to apply them to central questions in theory and practice. Of particular concern was the further development of freely available open source computer algebra systems and their interaction in order to create powerful new computational tools that transcend the boundaries of the individual disciplines involved.  The book covers a broad range of topics addressing the design and theoretical foundations, implementation and the successful application of algebraic algorithms in order to solve mathematical research problems. It offers a valuable resource for all researchers, from graduate students through established experts, who are interested in the computational aspects of algebra, geometry, and/or number theory.

Algorithms. --- Mathematics. --- Algebraic geometry. --- Commutative algebra. --- Commutative rings. --- Group theory. --- Number theory. --- Algebraic Geometry. --- Commutative Rings and Algebras. --- Group Theory and Generalizations. --- Number Theory. --- Math --- Science --- Number study --- Numbers, Theory of --- Algebra --- Groups, Theory of --- Substitutions (Mathematics) --- Rings (Algebra) --- Algebraic geometry --- Geometry --- Algorism --- Arithmetic --- Foundations --- Geometry, algebraic. --- Algebra. --- Mathematics --- Mathematical analysis

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This is the first in a series of three volumes dealing with important topics in algebra. It offers an introduction to the foundations of mathematics together with the fundamental algebraic structures, namely groups, rings, fields, and arithmetic. Intended as a text for undergraduate and graduate students of mathematics, it discusses all major topics in algebra with numerous motivating illustrations and exercises to enable readers to acquire a good understanding of the basic algebraic structures, which they can then use to find the exact or the most realistic solutions to their problems.

Mathematics. --- Associative rings. --- Rings (Algebra). --- Commutative algebra. --- Commutative rings. --- Algebra. --- Field theory (Physics). --- Group theory. --- Nonassociative rings. --- Number theory. --- Group Theory and Generalizations. --- Associative Rings and Algebras. --- Non-associative Rings and Algebras. --- Commutative Rings and Algebras. --- Field Theory and Polynomials. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Groups, Theory of --- Substitutions (Mathematics) --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis --- Rings (Algebra) --- Algebraic rings --- Ring theory --- Algebraic fields