First some notation. Let $F$ be a field, $E$ an algebraic extension of $F$ and $\overline{F}$ the algebraic closure of $F$. Let $\{E:F\}$ represents the number of non-zero homomorphisms from $E$ to $\overline{F}$ which leave the field $F$ fixed.
Suppose we have a tower of fields:
$F \subset E \subset K$
How can it be shown that $\{K:F\} = \{K:E\}*\{E:F\}$?
I know of a similar equality that deals with degrees of field extensions, but here we're talking about functions that go from $E$ to a much larger field $\overline{F}$ which makes it seem hard to visualize. What can I do to prove this?
As I recall, it depends on a strategy like this, modulo suitable care to the hypotheses:
If $k\subset L$ are fields and $\psi\colon L\to\Omega$, where $\psi$ is a field morphism and $\Omega$ is an algebraically closed field, then $\{L:k\}=\{\psi(L):\psi(k)\}$. This needs a proof, but it isn’t hard.