so I am currently studying separable extensions in John Fraleigh's abstract algebra book and have come upon this definiton of a separable field extension:
A finite extension $E$ of $F$ is a separable extension of $F$ if $\{E:F\}=[E:F]$. An element $\alpha$ of $\overline{F}$ is separable over $F$ if $F(\alpha)$ is a seperable extension of $F$. An irreducible polynomial $f(x)\in F[x]$ is separable over $F$ if every zero of $f(x)$ in $\overline{F}$ is separable over $F$.
I understand that this definition then leads us to study the multiplicity of the zeros of $\text{irr}(\alpha, F)$ and that an extension is only separable if the multiplicity of the zeros in $\overline{F}[x]$ is equal to one. However, I don't understand why this would lead us to call the extension separable. In my mind when I think of something that is separable I think of decomposing the structure in one way or another. Is there a way to think about seperable extensions in this manner, or am I completely missing something here?