I'm a little confused in regards to the definitions of normality and separability of field extensions in the context of Galois theory . The definitions seem very similar . In class they were defined as follows
Note: I'll give my interpretations using the example $x^3-5$ so it will be easier to spot any misunderstandings.
Separability:
Given a field $F$
i) We say $f(x)\in F[x]$ is separable over $F$ if each irreducible factor of $f$ has simple roots in a splitting field.
My interpretation: $f(x)=x^3-5$ has 3 distinct roots $\sqrt[3]{5}, w \sqrt[3]{5}, w^2 \sqrt[3]{5}$. As they are distinct, i.e. appear once, they are simple roots in some splitting field $E$, so $f(x)$ is separable.
ii) Given $E/F$ and $\alpha \in E$, algebraic over $F$, we say $\alpha$ is separable over $F$ if it is the root of a separable polynomial over $F$.
My interpretation: As in (i) I gave reasons to why I thought $x^3-5$ is separable, then we know each root $\alpha$ is separable.
iii) We say an algebraic extension $E/F$ is separable if each $\alpha \in E$ is separable over $F$.
My interpretation : The splitting field of $x^3-5$ is $E(\sqrt[3]{5},w)$, by ii) $\sqrt[3]{5}$ is separable. For $w$ this is a root of $x^3-1=(x-1)(x^2+x+1)$. It can't be equal to $1$, therefore it's a root of $g(x)=x^2+x+1$. This equation has distinct roots $w,w^2$, so they are simple in any splitting field, so $g(x)$ is separable over $F$. Using this with ii) and iii), we see that $E/\Bbb Q$ is separable.
Normality :
For $E/F$ if $m(x) \in F[x]$ is irreducible and $m$ has a root in $E$, then $m$ splits over E.
My interpretation: $E/\Bbb Q$ is normal as $m(x)=x^3-5$ is the minimum polynomial. containing the roots $\sqrt[3]{5}, w \sqrt[3]{5}, w^2 \sqrt[3]{5}$ is irreducible over $\Bbb Q$, $m$ has a root in $E$ and splits in $E$.
Additional note :So to my understanding separability of an extension is basically saying each element of $E$ is the root of some polynomial which can be split over E, while normality is saying all minimum irreducible polynomials which have roots in $E$ split over $E$. These seem like the same statements in reverse except for the additional caveat of minimum polynomial in the normality definition. Could someone explain the deeper difference of these two concepts to me please?
Edit: There is a link to an answer but I do not feel it is a duplicate to mine for the following reasons :It certainly asks the same thing, but I feel like my question is much more specific , at least in the part where I gave my interpretation . Also in the answer of the linked question he talks about the characteristic of the field , I was trying to understand he concept without using this. Additionally I had a different concrete example than the ones used in the answer so the reasoning is different. Although I see how the linked question is similar to mine in an over arching way, I feel like the details of the two are sufficiently different to say that they are not duplicates.