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Principal Ideal Domains
Principal Ideal Domains

... The note above immediately proves the following result. Theorem 57. If a and b are nonzero elements in the commutative ring R such that (a, b) = (d), then d = gcd(a, b). Note 58. It is important to point out that the theorem above is giving us a sufficient condition, but it is not necessary. For exa ...
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Math 323. Midterm Exam. February 27, 2014. Time: 75 minutes. (1

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Advanced Algebra II Notes 7.1 Polynomial Degree and Finite

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WHAT IS A POLYNOMIAL? 1. A Construction of the Complex

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17. Field of fractions The rational numbers Q are constructed from

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Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator. Many important conjectures involving polynomial rings, such as Serre's problem, have influenced the study of other rings, and have influenced even the definition of other rings, such as group rings and rings of formal power series.A closely related notion is that of the ring of polynomial functions on a vector space.
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