# Field theory: an equality involving the number of homomorphisms from an extension $E$ of $F$ to $\overline{F}$

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?

• "The closure"?? The algebraic, the separable, the normal...what closure? May 18 '16 at 17:02
• The field in which every polynomial with coeff's in F splits, the algebraic closure. May 18 '16 at 17:05