Difference between normal extension and splitting extension

Question

Can you give me an example which shows well the difference between them?

Answer 1

We have the theorem: let be $[K,k]<\infty $ then the following is equivalent:

$1)$ $K$ is normal extension

$2)$ $K$ is splitting field over $k$ for some $f(x)\in k[x]$.

but we know that if $\bar k$ is algebraically closed field of $k$ then $\bar k$ is normal extension of $k$ but $\bar k$ isnot splitting field of $k$.

for example: if $f(x)=x^2+1 \in \mathbb Q[x] $ then $\mathbb Q(i)$ is spilitting field of $f(x)$ and $\mathbb Q(i)$ is normal extension of $\mathbb Q$. on other hand we have $\bar{\mathbb Q}$ is normal extension but not spilitting field.