$[K:F_1]=[K:F_2]$, are $F_1, F_2$ isomorphic?

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$K$ is a field extension of field $F_1$, $F_2$, all of them are finite fields. There is no other conditions. I want to know is the assertion possible?

Could you please give a brief explanation?

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There's only one finite field of a given order, so yes.