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- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/02%3A_Fields_and_RingsIn this chapter we will set the ground work for answering that question by introducing ideas that will assist us in streamlining our investigation into factorization.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/zz%3A_Back_Matter/20%3A_GlossaryExample and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pag...Example and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages] (Optional) Caption for Image (Optional) External or Internal Link (Optional) Source for Definition "Genetic, Hereditary, DNA ...") (Eg. "Relating to genes or heredity") The infamous double helix CC-BY-SA; Delmar Larsen Glossary Entries Definition Image Sample Word 1 Sample Definition 1
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/02%3A_Fields_and_Rings/2.04%3A_Principal_Ideals_and_Euclidean_DomainsIn this section, we begin a set-theoretic structural exploration of the notion of ring by considering a particularly important class of subring which will be integral to our understanding of factoriza...In this section, we begin a set-theoretic structural exploration of the notion of ring by considering a particularly important class of subring which will be integral to our understanding of factorization.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/02%3A_Fields_and_Rings/2.03%3A_Divisibility_in_Integral_DomainsWhen we introduced the notion of integral domain, we said that part of the reason for the definition was to capture some of the most essential properties of the integers. This is the heart of abstrac...When we introduced the notion of integral domain, we said that part of the reason for the definition was to capture some of the most essential properties of the integers. This is the heart of abstraction and generalization in mathematics: to distill the important properties of our objects of interest and explore the consequences of those properties. One such important property of Z is cancellation.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/00%3A_Front_Matter
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/03%3A_FactorizationIn this chapter, we come to the heart of the text: a structural investigation of unique factorization in the familiar contexts of Z and F[x].
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/01%3A_The_Integers/1.01%3A_Induction_and_Well-OrderingIn 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 ...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.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/04%3A_Ideals_and_Homomorphisms_and_testThe culmination is the result that any Euclidean domain is a unique factorization domain; that is, in an integral domain with a well-behaved division algorithm, a nonzero nonunit necessarily factors u...The culmination is the result that any Euclidean domain is a unique factorization domain; that is, in an integral domain with a well-behaved division algorithm, a nonzero nonunit necessarily factors uniquely into irreducibles. In order to expediently develop that result, we ignored many concepts that are otherwise interesting and useful in a first course in abstract algebra. This chapter is a coda that seeks to fill in some of those gaps.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/03%3A_Factorization/3.02%3A_Factorization_in_Euclidean_DomainsIn this section, our explorations of the structural arithmetic properties that guarantee unique factorization culminate in Theorem 3.2.7 . Specifically, we'll see that all Euclidean domains possess t...In this section, our explorations of the structural arithmetic properties that guarantee unique factorization culminate in Theorem 3.2.7 . Specifically, we'll see that all Euclidean domains possess the unique factorization property. To prove this theorem, we will rely in part on an interesting property of chains of ideals in Euclidean domains.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)One of the concepts fundamental to the historical development of algebra is the notion of factorization; closely related questions that have driven the development of algebra over the centuries are: w...One of the concepts fundamental to the historical development of algebra is the notion of factorization; closely related questions that have driven the development of algebra over the centuries are: when does a polynomial equation have solutions in a particular number system, and is there a systematic way to find them? The goal of this book is to explore the idea of factorization from an abstract perspective.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Rings_with_Inquiry_(Janssen_and_Lindsey)/01%3A_The_Integers/1.02%3A_Divisibility_and_GCDs_in_the_IntegersIn this section, we begin to explore some of the arithmetic and algebraic properties of Z.
