While reading a theorem on Galois theory I came up with this statement "Let $E/F$ be a finite separable extension. Then $E$ is contained in an extension of $K$ which is Galois over $F$.
Now my question is about first statement. Because according as what I know if$E/F$ is a finite extension then it's an algebraic extension and as it's separable hence it's a normal extension also. So $E/F$ is normal separable extension, hence it is Galois. So what am I thinking wrong. It will be really great if you can help me with it