Factorization in Integral Domains and in Polynomial Rings
Chapter 6 extends to rings the concepts of divisibility, greatest common divisor, least common multiple, division algorithm, and Fundamental Theorem of Arithmetic for integers with the help of theory of ideals. The main aim of this chapter is to study the
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Basic Modern Algebra with Applications
Basic Modern Algebra with Applications
Mahima Ranjan Adhikari r Avishek Adhikari
Basic Modern Algebra with Applications
Mahima Ranjan Adhikari Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC) Kolkata, West Bengal, India
Avishek Adhikari Department of Pure Mathematics University of Calcutta Kolkata, West Bengal, India
ISBN 978-81-322-1598-1 ISBN 978-81-322-1599-8 (eBook) DOI 10.1007/978-81-322-1599-8 Springer New Delhi Heidelberg New York Dordrecht London Library of Congress Control Number: 2013954442 © Springer India 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated to NABA KUMAR ADHIKARI (1912–1996), a great teacher, on occasion of his birth centenary
Preface
This book is designed to serve as a basic text of modern algebra at the undergraduate level. Modern mathematics facilitates unification of different areas of mathematics. It is characterized by its emphasis on the systematic study of a number of abstract mathematical structures. Modern algebra provides a language for almost all disciplines in contemporary mathematics. This book introduces the basic language of modern algebra through a study of groups, group actions, rings, fields, vector spaces, modu
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