How does one go about inductively proving that
If $K_0 \subseteq K_1 \subseteq \cdots \subseteq K_n$ are subfields of the complex numbers, then is the following true?$$[K_n : K_1 : K_0] = [K_n : K_{n - 1}][K_{n - 1} : K_{n - 2}] \cdots [K_1 : K_0]$$
I'd imagine that the base case is $K_0 \subseteq K_1$?
Thank you so much!
We shall proof by mathematical induction True for n=1 [K1:k0] = [K1:k0] Suppose true for n=t [Kt:k0]=[kt:kt-1][kt-1:kt-2]...[k1:k0] (1)
We have to proof for n =t+1 That is [Kt+1:k0]=[kt+1:kt][kt:kt-1][kt-1:kt-2]...[k1:k0] Now use (1) [Kt+1:k0]=[kt+1:kt][kt:k0] [Kt+1:k0]=[kt+1:k0] by short tower law Hence proved