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

  1. Sep 13, 2014 #1
    The problem statement, all variables and given/known data

    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.
     
  2. jcsd
  3. Sep 16, 2014 #2
    I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
     
  4. Sep 17, 2014 #3

    jbunniii

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    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?
     
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