Is there concrete example of finite extension which is not normal?

Question

I'm confusing about finite extension and normal extension. $E/F$be a finite extension. Then $E=F (a_{1},a_{2},a_{3},....,a_{n})$ and each $a_{i}$ is algebraic over $F$ Then there exist $f_{i}(x) $ in $F [x] $ which has $a_{i} $ as a root. Then I think $E$ is splitting field of product of all $f_{i}(x) $s. If it is right, there is no difference between finite extension and normal extension. Is there concret example of finite extension which is not normal?

Answer 1

Your reasoning doesn't make sense. The roots certainly lie in the splitting field, but why should you the splitting field be the same as the original? $\def\qq{\mathbb{Q}}$$\qq(\sqrt[3]{2}) / \qq$ is certainly a finite extension but not normal since it does not have the non-real roots of $x \mapsto x^3-2$.