Description

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This is the classic text for the first undergraduate course in abstract algebra; it follows the "rings before groups" philosophy that relies on the student's familiarity with the ring of integers to make the abstract axioms feel more concrete. The text and exercises are coordinated to help the student learn to read and write mathematical proofs. To achieve an understanding of the methods of proof, the unique factorization theorems are presented several times in different contexts. The integers are covered first, then polynomial rings over a field and, in the final chapter, factorization in principal ideal domains. Results on finite fields up through existence and uniqueness are proved without the use of linear algebra. The material on groups is studied in three chapters, the first giving definitions and examples, including permutation groups; the second gives the classification of finite abelian groups; the third discusses groups acting on sets and gives the proofs of the Sylow theorems. There are many worked examples and a large number of exercises of varying levels of difficulty. By carefully selecting sections to be covered in class lectures, an instructor will find this text suitable for almost any syllabus giving an introduction to abstract algebra.