A_First_Course_in_Algebra-Fraleigh-7e-solutions.pdf

Instructor’s

SolutionsManual

toaccompany

AFirstCoursein

AbstractAlgebra

SeventhEdition

JohnB.Fraleigh

UniversityofRhodeIsland

Preface

This manual contains solutions to all exercises in the text, except those odd-numbered exercises for which

fairly lengthy complete solutions are given in the answers at the back of the text. Then reference is simply

given to the text answers to save typing.

I prepared these solutions myself. While I tried to be accurate, there are sure to be the inevitable

mistakes and typos. An author reading proof rends to see what he or she wants to see. However, the

instructor should ﬁnd this manual adequate for the purpose for which it is intended.

Morgan, Vermont

J.B.F

July, 2002

CONTENTS

0. Sets and Relations 1

I.GroupsandSubgroups

1. Introduction and Examples 4

2. Binary Operations 7

3. Isomorphic Binary Structures 9

4. Groups 13

5. Subgroups 17

6. Cyclic Groups 21

7. Generators and Cayley Digraphs 24

II.Permutations,Cosets,andDirectProducts

8. Groups of Permutations 26

9. Orbits, Cycles, and the Alternating Groups 30

10. Cosets and the Theorem of Lagrange 34

11. Direct Products and Finitely Generated Abelian Groups 37

12. Plane Isometries 42

III.HomomorphismsandFactorGroups

13. Homomorphisms 44

14. Factor Groups 49

15. Factor-Group Computations and Simple Groups 53

16. Group Action on a Set 58

17. Applications of G-Sets to Counting 61

IV.RingsandFields

18. Rings and Fields 63

19. Integral Domains 68

20. Fermat’s and Euler’s Theorems 72

21. The Field of Quotients of an Integral Domain 74

22. Rings of Polynomials 76

23. Factorization of Polynomials over a Field 79

24. Noncommutative Examples 85

25. Ordered Rings and Fields 87

V.IdealsandFactorRings

26. Homomorphisms and Factor Rings 89

27. Prime and Maximal Ideals 94

28. Grobner Bases for Ideals 99

iii

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