Extensions of Rings and Fields
A given ring may fail to have certain properties that are necessary for solving a particular problem. However, it may be possible to construct a larger ring that has the required properties. Thus, for example, there exist equations of the form ax = b, a ≠0 that have no solutions in the domain of integers. The field of rational numbers is constructed for the purpose of insuring the solvability of equations of this type. The method used to construct this extension can be generalized so as to apply to any commutative integral domain. This type of extension is one of those that we consider in this chapter. Among others we define also rings of polynomials, field extensions and rings of functions. We derive some of the properties of these extensions and, in particular, we determine the algebraic structure of any field.
KeywordsIntegral Domain Polynomial Ring Homomorphic Image Principal Ideal Symmetric Polynomial
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