Equivalent definitions of Galois extension.

Question

I am studying Galois theory.I encountered some equivalent definitions of Galois extension:

The following are equivalent for a finite extension:

$1. E/K$ is a splitting field of separable polynomial over $K$.

$2. E^{\operatorname{Gal}(E/K)}=K$ for some group of automorphisms of $E$.

$3.E/K$ is normal and separable.

$4. [E:K]=|Gal(E/K)|$.

I am unable to prove the equivalence among these definitions.I am new in Galois theory,it would be nice if someone provides hint for proving these equivalences.Can someone help me?For starter,I am looking for a direct proof of $(1)\implies (2)$.