List all the ideals of Q[x] containing the element f(x) = (x2 + x - 1)3(x-3)2

[phyguy321]

Homework Statement

List all the ideals of Q[x] containing the element
f(x) = (x² + x - 1)³(x-3)²

Homework Equations

The Attempt at a Solution

Why would an ideal contain this element?
when all ideals in Q[x] are defined by being divisible by (x-2)

 

[enigmahunter]

The smallest ideal containing above f(x) is probably (f(x)).
The maximal ideal containing f(x) is both (x^2+x-1) and (x-3).

You can find the ideals in between.

 

[HallsofIvy]

enigmahunter, do you mean, by (f(x)) the ideal generated by f(x): the set of all polynomials of the form af(x) where a is a rational number? And, I presume that, by "(x^2+x-1) and (x-3)" you mean the ideal generated by those two polynomials.

phyguy321, where did you get the idea that "all ideals in Q[x] are defined by being divisible by (x-2)"? The set of all polynomials, p(x) with rational coefficients, such that p(1)= 0, for example, is an ideal in Q[x]. That includes many polynomials that are NOT divisible by x-2.

 

[enigmahunter]

(f(x)) or <f(x)> (which one is standard?) denotes the ideal of Q[x] (polynomials with coefficients Q) generated by a subset of Q[x], which is f(x).
It is the set of all elements whose form are r*f(x) where r belongs to Q[x].