I tried to solve this problem, but I'm not sure if my solution is correct.
The problem is the following one:
Let $~K~$ be a field, $~f(x)~$ and $~g(x)~$ two irreducible polynomials in $~K[x]~$ with the same grade $~d \geq 2~$. Determine (both for $~d=2~$ and for a general value of $~d~$) the implications between these two proprieties:
$~1.~~$ the ring $~\frac{K[x,y]}{(f(x),g(y)}~$ is a field.
$~2.~~$ the extensions $~\frac{K[x]}{f(x)}~$ and $~\frac{K[x]}{g(x)}~$ of $~K~$ are not isomorphic
I found an example which proves that $~2~$ doesn't imply $~1: ~~~K=Q, ~~~f(x)=x^{2}-2, ~~~g(x)=x^{2}~$
However I wasn't able to find the other implication. Supposing $~1\to 2~$ true I tried to proof this implication using the negation of this sentence, but I wasn't able to get a conclusion.
Is there a simpler way to prove if the implication $~1\to 2~$ is true or not?