Introduction to Modern Algebra

This text is intended for use in a first course in abstract algebra. In the present undergraduate curriculum, the first course often serves several purposes. The student is expected to learn a collection of facts, some methods and ideas, and to learn to read and write correct proofs. All of these requirements have been carefully considered in the preparation of this text. The material in Chapter 4, for example,presents the unique factorization theorem, with all the required preliminaries, first in the familiar setting of the integers, then in the less familiar but still concrete setting of the ring of polynomials over a field, and then finally in the very general situation of a Euclidean domain. There is considerable information contained in the exer- cises. Some important theorems are left to the exercises due to lack of time and space. The highly motivated student should be encouraged to read all the exercises even though it is not reasonable to assign all of them to be worked. There are some exercises scattered throughout the text deal- ing with two-by-two matrices. Even though no formal theory of matrices is presented in the text, the student should have no serious difficulty with these if the exercises on this topic are worked from the beginning. A number of topics that might be called "enrichment" material are marked with an asterisk (*) in the Contents. These would not ordinarily be covered in lectures but may be used for special projects, or as reading assignments for motivated students.