Proving f(x) Divides g(x) iff g(x) in <f(x)>

[tinynerdi]

Homework Statement

let F be a field and f(x),g(x) in F[x]. Show that f(x) divides g(x) if and only if g(x) in <f(x)>

Homework Equations

let E be the field F[x]/<f(x)>

The Attempt at a Solution

<=> if f(x) divides g(x) then g(x) in <f(x)>
Proof: Suppose f(x) divides g(x)q(x). then g(x)q(x) in <f(x)>. which is maximal. Therefore <f(x)> is a prime ideal. Hence g(x)q(x) in <f(x)>. implies that either g(x) in <f(x)> giving f(x) divides g(x) or that q(x) in <f(x)> giving f(x) divides q(x). But we want that g(x) in <f(x)> giving f(x) divides g(x).

can this prove go both way if it is right?

 

[mrbohn1]

I think this is even simpler than you think. The condition for g(x) to divide f(x) is that there is q(x) in F[x] such that f(x)=q(x)g(x), and this is exactly the condition for g(x) to belong to the ideal generated by f(x).