As children we start exploring the properties and structure of the positive integers as soon as we learn to count and we extend our understanding throughout our schooling as we learn about new operations and collections of numbers. We begin our journey into abstract algebra with an overview of some familiar (and some possibly unfamiliar) properties of the integers that are relevant to our course of inquiry. With this foundation set, we will see in later chapters just how far we can extend these properties in more abstract setting.
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1.1: Induction and Well-Ordering
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In this section we will assume the basic algebraic/arithmetic properties of the integers such as closure under addition, subtraction, and multiplication, most of which we will formalize via axioms in subsequent sections.
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1.2: Divisibility and GCDs in the Integers
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In this section, we begin to explore some of the arithmetic and algebraic properties of Z.
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1.3: Primes and Factorization
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As described in the Introduction, our main goal is to build a deep structural understanding of the notion of factorization.
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1.4: The Integers modulo m
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The foundation for our exploration of abstract algebra is nearly complete. We need the basics of one more "number system" in order to appreciate the abstract approach developed in subsequent chapters.