The last few days I read up on étale morphisms of rings / affine schemes and the étale fundamental group. I have two closely connected questions.
For $K$ a field a ring-homomorphism $K \rightarrow A$ is étale if and only if $A\cong L_1 \times ... \times L_r$ as $K$-algebras, where all $L_i \mid K$ are finite separable field extensions. It is often stated that the étale fundamental group of a field is it's absolute Galois-group $\operatorname{Gal}(K^s,K)$, where $K^s$ is the separable closure. I don't quite see how we can go from a product of finite separable extensions to single finite separable extensions, ie. why $$\pi_1^\text{et}(K) = \lim \operatorname{Aut}_K(L_1 \times \dots \times L_r) \;\;\text{and}\;\; \operatorname{Gal}(K^s,K) = \lim \operatorname{Aut}_K(L)$$ are supposed to give the same result, especially, since a product of finite separable extensions may not admit a
morphismmonomorphism into a common separable extension. Edit:There exists morphisms though.Why is a finite field $F$ considered to be an étale Eilenberg-Maclane space?
When the issues in 1 are resolved,I see why $\pi_1^\text{et}(F) = \widehat{\Bbb Z}$, but just for spaces this shouldn't be the only requirement. I tried to look for more "evidence", but it seems like higher étale fundamental groups are quite hard to define. So is there some other naive reason, one could say that?
As always, thank you for your time and efforts!