I know that $K'\cdot K''$ is an unramified extension of $K$ but I don't know why $K'\cdot K''$ have a residue field $k'$.
is it always true that $K_1\cdot K_2$ have a residue field $k_1 \cdot k_2$? (where $k_1,k_2$ are residue fields of $K_1, K_2$)
I think that if we prove the proposition 7.50 , then we can use " $K_1\cdot K_2$ have a residue field $k_1 \cdot k_2$" in this situation.
However, we can't use that fact while proving this proposition.
How can I prove this?
Thank you for your attention.
reference(J.S. Milne's Algebraic Number Theory) and this post 1: Strange reasoning of unramified extensions having same residue fields are the same.