EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS Mahmut Kuzucuoğlu Middle East Technical University matmah@metu.edu.tr Ankara, TURKEY April 18, 2012 ii iii TABLE OF CONTENTS CHAPTERS 0. PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1. SETS, INTEGERS, FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. GROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. RINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4. FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5. INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 iv v Preface These notes are prepared in 1991 when we gave the abstract algebra course. Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. I would like to thank Bülent Büyükbozkırlı for his help during the preparation of these notes. I would like to thank also Prof. İsmail Ş. Güloğlu for checking some of the solutions. Of course the remaining errors belongs to me. If you find any errors, I should be grateful to hear from you. Finally I would like to thank Aynur Bora and Güldane Gümüş for their typing the manuscript in LATEX. Mahmut Kuzucuoğlu I would like to thank our graduate students Tuğba Aslan, Büşra Çınar, Fuat Erdem and İrfan Kadıköylü for reading the old version and pointing out some misprints. With their encouragement I have made the changes in the shape, namely I put the answers right after the questions. 20, December 2011 vi M. Kuzucuoğlu 1. SETS, INTEGERS, FUNCTIONS 1.1. If A is a finite set having n elements, prove that A has exactly 2 distinct subsets. n Solution: Apply Induction on n.