Kunze Linear Algebra · Problem 4.4.1

Polynomials

Mathematics — Methods · ideals-in-polynomial-ring

Problem

Let QQ be the field of rational numbers. Determine which of the following subsets of Q[x]Q[x] are ideals. When the set is an ideal, find its monic generator. ((a)) all ff of even degree; ((b)) all ff of degree ≥5\geq 5; ((c)) all ff such that f(0)=0f(0) = 0; ((d)) all ff such that f(2)=f(4)=0f(2) = f(4) = 0; ((e)) all ff in the range of the linear operator TT defined by T(∑i=0ncixi)=∑i=0ncii+1xi+1.T\left(\sum_{i=0}^n c_i x^i\right) = \sum_{i=0}^n \frac{c_i}{i+1} x^{i+1}.

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