I have got no clue of how to start with the problem .Now I know that an ideal is a kernel of a ring homorphism but why would it be a principal ideal in $F[x]$.What is the intuition behind this ?(I would like to know why is it happening instead of a formal proof)
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4Does this answer your question? If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field? – José Carlos Santos Mar 01 '20 at 15:29
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It is because $(f,g)=(\gcd(f,g))$ and the RHS ideal is larger than $(f)$ iff $\gcd(f,g)$ has lower degree than $f$ – reuns Mar 01 '20 at 15:33