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The tensor product $F_1 \otimes F_2$$F_1 \otimes_F F_2$ is not 0, hence it has a quotient which is a field. This contains the images of both $F_i$.
The tensor product $F_1 \otimes F_2$ is not 0, hence it has a quotient which is a field. This contains the images of both $F_i$.
The tensor product $F_1 \otimes_F F_2$ is not 0, hence it has a quotient which is a field. This contains the images of both $F_i$.
