What is the relation between normal extension and separable extension?
Let F be the algebraic extension of K, if F is a separable extension of K,if and only if F is a normal extension of K? is this correct?


Neither implies the other. So, there exist separable extensions that are not normal, and normal extensions that are not separable.

$\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is a separable extension that is not normal. It is separable because any extension of characteristic zero fields is separable, but it is not normal because (for example) not all algebraic conjugates of $\sqrt[3]{2}$ lie in the field $\mathbb{Q}(\sqrt[3]{2})$ (its conjugates are $\zeta_3\sqrt[3]{2}$ and $\zeta_3^2\sqrt[3]{2}$ where $\zeta_3$ is a cube root of unity; these are complex numbers, while $\mathbb{Q}(\sqrt[3]{2})\subset\mathbb{R}$).

$\mathbb{F}_p(\sqrt[p]{T})/\mathbb{F}_p(T)$ is a normal extension that is not separable. It is normal because $\mathbb{F}_p(\sqrt[p]{T})$ is the splitting field of $x^p-T\in F_p(T)[x]$, but it is not separable because the minimal polynomial of $\sqrt[p]{T}$ is $x^p-T=(x-\sqrt[p]{T})^p$ which does not have distinct roots.

A field extension $L/K$ that is both normal and separable is called a Galois extension.


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