(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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)>

2. Relevant equations

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

3. 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?

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# Homework Help: Factor ring

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