The ideals of ##\mathbb Q[X]##

[mahler1]

Homework Statement

Find all the ideals of $\mathbb Q[X]$

The attempt at a solution

Suppose $I \subset \mathbb Q[X]$ is an ideal with an element $p(x) \neq 0$. Since $\mathbb Q[X]$ is an euclidean domain (the function $degree(f)$ is an euclidean function), then $\mathbb Q[X]$ is a principal ideal domain. This means that $I$ can be generated by an element. Now, since $I$ is an ideal, in particular is a subgroup under addition, so $-p(x), np(x) \in I$ for $n \in \mathbb Z$.

I am not so sure what to do next. I got stuck here, any help or suggestions would be appreciated.

 

[Greg Bernhardt]

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

 

[jbunniii]

As you noted, $\mathbb{Q}[x]$ is a principal ideal domain. Therefore every ideal is of the form $(p(x))$ for some polynomial $p(x) \in \mathbb{Q}[x]$.

So the question is how to identify when two polynomials $p(x)$ and $q(x)$ generate the same ideal. Suppose that $p(x) \in (q(x))$ and $q(x) \in (p(x))$. This means that $p(x) = q(x)r(x)$ and $q(x) = p(x)s(x)$ for some polynomials $r(x)$ and $s(x)$. Then $p(x) = p(x)s(x)r(x)$, which we can rearrange as
$$p(x)(1 - s(x)r(x)) = 0$$
Keeping in mind that we are working in a domain, what can you conclude from this?