Rings and Polynomials Textbook Page

15 Rings and Polynomials —Sauron Ring theory has many applications in computing, counting, cryptography, and communications. Rings are sets with two operations, “+” and “·” (usually called addition and multiplication, respectively). In many ways rings are like vector spaces, and much of our treatment of the theory of rings will mirror our treatment of vector spaces at the beginning of this book. The polynomial ring is the set ℤ with addition and multiplication as the two operations. But there are many other interesting and useful examples of rings, including ℱₙ, ℂ, ℤ[√−1], and ℤ[X] for any normed linear space X. Ring theory shows that many results that are true for the integers also apply to these rings. We begin with a survey of the basic properties of rings, with a special focus on the similarities between rings and vector spaces. We then narrow our focus to a special class of rings called Euclidean domains, and to quotients of Euclidean domains. These include rings of polynomials in one variable and the ring of all matrices that can be formed by applying polynomials to one given matrix. A Euclidean domain is a ring in which the Euclidean algorithm holds, and we show in the third section that this also implies the fundamental theorem of arithmetic (unique prime factorization) holds in these rings. Applying this to the ring of polynomials, we get an alternative form of the fundamental theorem of algebra. The fourth section treats two great homomorphisms, which are the ring-theoretic

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