Wikipedia's equivalent definitions of finite separable field extensions

Question

According to Wikipedia, given a finite field extension $E\supseteq F$, the following are equivalent:

1. $E$ is separable over $F$ i.e. for any $\alpha\in E$ the minimal polynomial of $\alpha$ over $F$ is separable.
2. $E=F(a_1,\ldots,a_r)$ where $a_1,\ldots,a_r$ are separable elements of $E$.
3. $E=F(a)$ where $a$ is a separable element of $E$.
4. If $K$ is an algebraic closure of $F$, then there are exactly $[E:F]$ field homomorphisms of $E$ into $K$ which fix $F$.
5. If $K$ is a normal extension of $F$ containing $E$, then there are precisely $[E:F]$ field homomorphisms of $E$ into $K$ which fix $F$.

The first three definitions are equivalent by the Primitive Element Theorem, and naturally $5\implies 4$. However, I fail to see how $4$ and $5$ are equivalent to $1, 2, 3$ (even when thinking of the Fundamental Theorem of Galois Theory).

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Why are definitions 4 and 5 above equivalent to the rest?

It would be sufficient to show $4$ implies either $1$,$2$, or $3$.