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I know that I am looking for a counterexample, the statement is not true when the Disc(K) and the Disc(L) are not coprime.

I have been trying to use $Disc(K)=[\mathcal{O}_K:K]^2Disc(\mathcal{O}_K)$, but am not having much success.

I have particularly been looking at the two fields generated by $f(x)=x^2-3$ and $g(x)=x^2-2$.

Any help would be greatly appreciated I have been trying to do this for ages.

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