16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 4.4.1

Polynomials

Mathematics — Linear Algebra · polynomial-ideals

Problem

Let

Q

be the field of rational numbers. Determine which of the following subsets of

Q[x]

are ideals. When the set is an ideal, find its monic generator. ((a)) all

f

of even degree; ((b)) all

f

of degree

\geq 5

; ((c)) all

f

such that

f(0) = 0

; ((d)) all

f

such that

f(2) = f(4) = 0

; ((e)) all

f

in the range of the linear operator

T

defined by

T\!\left(\sum_{i=0}^n c_i x^i\right) = \sum_{i=0}^n \frac{c_i}{i+1}\, x^{i+1}.