### Simple Extensions with the Minimum Degree Relations of Integral Domains

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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. The field of rational numbers is constructed for the purpose of insuring the solvability of equations of this type.

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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. Viewed 3k times.

Thank you. Adam Hughes What is true is that every element of the extension is a linear combination with coefficients in the base field of powers of the generator. I'll go back and fix that.

Keenan Kidwell Keenan Kidwell My book explained it poorly. Adam Hughes Adam Hughes Thanks guys! Since the imaginary unit is the generator shouldn't powers of it create all the complex numbers? I know from elementary math that makes no sense.

But from the definition of generator it should. I thought this meant the whole field including the additive identity were cyclic under the the generator, in this case the imaginary unit.

Sign up or log in Sign up using Google. Introduction to Abstract Algebra.

This is followed by a range of current topics such as multilinear algebra, matrix polynomials and equations, perturbation theory, pseudospectra, inverse problems, integral domains , and spectral sets, among many others. Handbook of Linear Algebra, 2nd Edition. Simple extensions with the minimum degree relations of integral domains.

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They then discuss integers, rings and ordered integral domains. Among the topics are greatest common divisors, integer multiples and exponents, quotients of polynomial rings, divisibility and factorization in integral domains , subgroups of cyclic groups, cosets and Lagrange's theorem, the fundamental theorem of finite abelian groups, and check digits.