I'm reading through some Dummit and Foote chapters on Galois theory and came across this proposition.
Proposition 21. Let $K_{1}$ and $K_{2}$ be Galois extensions of a field $F$. Then
(1) The intersection $K_{1} \cap K_{2}$ is Galois over F
(2) The composite $K_{1}K_{2}$ is Galois over $F$. The Galois group is isomorphic to the subgroup
$H = \{(\sigma, \tau) | \sigma|_{K_{1}\cap K_{2}} = \tau|_{K_{1} \cap K_{2}}\}$ of
of the direct product of $Gal(K_{1}/F) \times Gal(K_{2}/F)$
That subgroup $H$ seems very much like a collection of germs to me, is there some relation between Galois groups and sheaves?