Abstract
Let an ideal I ∈ℂ {x} be given. In this chapter we will define and study standard bases of the ideal I. The main idea here is that one wants to have a “good” representative for any coset of I. In fact, the Weierstraß Division Theorem provides us with the simplest example of a standard basis. Consider f ∈ ℂ {x} which is regular of order b in x n . For any g ∈ ℂ {x} we have a unique representation g = qf + r with r a polynomial of degree less than b. From the uniqueness statement in the Weierstraß Division Theorem, it follows without difficulty that the remainder r only depends on the coset g + (f) of (f). Of course, the remainder r depends on the choice of x n .KeywordsNormal FormStandard BasisCauchy SequencePolynomial RingFormal Power SeriesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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