I just begun to learn algebraic extension. In some notes, I see two examples.
Example 1. The polynomial $X^3-2$ has one real root $\sqrt[3]2$ and two non real roots in $\mathbb{C}$. Therefore, the extension $\mathbb{Q}[\sqrt[3]2]/\mathbb{Q}$ (which is separable) is not normal.
How can I show it is separable and not normal?
Example 2. The extension $\mathbb{F}_p(T)/\mathbb{F}_p(T^p)$ (which is normal) is not separable because the minimal polynomial of $T$ is not separable.
How can I show it is normal but not separable?
An extension $E/F$ is separable if for every element $\alpha\in E$, the minimal polynomial of $\alpha$ over $F$ has no multiple roots. A test for multiplicities of roots can be carried out by putting in the roots in the formal derivative of the polynomial. More concretely, if $\{\alpha_1,...,\alpha_n\}$ are all the roots of $f(x)$, then $f(x)$ has no multiple roots if and only if $f'(\alpha_i)\neq 0$ for all $i$. In the case of an extension over $\mathbb Q$, or more generally over a field of charactersitic $0$, any extension is separable. This follows from the fact that any irreducible polynomial over $\mathbb Q$ is separable. If $f(x)$ is irreducible and contains a multiple root $\alpha$, then $\alpha$ is also a root of $f'(x)$, hence $f$ divides $f'$ by irreducibility. This is a contradiction since $f'$ is not identically $0$, so it has positive but strictly smaller degree than $f$.
The field extension $\mathbb Q(2^{\frac13})/\mathbb Q$ is not normal because it does not contain all the roots to the polynomial $x^3-2$. As $\mathbb Q(2^{\frac 13})$ is a real field, so it cannot contain the non-real roots.
For the second extension, the minimal polynomial of $T$ is given by $x^p-T^p\in\mathbb F_p(T^p)[x]$. It has multiple roots because in characteristic $p$, $x^p-T^p=(x-T)^p$ in $\mathbb F_p(T)$. (You can prove that this polynomial is irreducible by showing that $(x-T)^k\notin\mathbb F_p(T^p)[x]$ for any $k<p$.) Since this polynomial is non-separable, the field extension is not separable. To see that it is normal, simply notice that $\mathbb F_p(T)=\mathbb F_p(T^p)(T)$, i.e. the field extension is obtained by adjoining the single root $T$. But by the above discussion, $T$ is the only root of the minimal polynomial of $T$. So it is in fact a splitting field, which is normal.