I'm reading Fraleigh's A First Course in Abstract Algebra, Seventh Edition, and he makes the following definition on page 448:
A finite extension $K$ of $F$ is a finite normal extension of $F$ if $K$ is a separable splitting field over $F$.
He never explicitly defined what a normal extension is, but I've gathered from here that $K$ is a normal extension of $F$ if $K$ is a splitting field over $F$. Now, I'm trying to figure out how this relates to Fraleigh's definition of a "finite normal extension." Does his definition mean that if $K$ is a finite extension and a splitting field over $F$, then $K$ is separable over $F$? Is that even true? Please help clarify the terminology.
It seems to me like the only way to marry the various definitions would be to define a "normal extension of $F$" as a separable splitting field over $F$.
Best Answer
As Andreas Caranti pointed out, there do exist finite extensions that are splitting fields and yet are not separable. I think now that the best way to make sense of the terminology is to abandon the idea of a "finite normal extension." Leave the definition of a normal extension $E$ of $F$ as a splitting field over $F$, then based on this, define a Galois extension $E$ of $F$ as an extension that is both normal and separable. Then, we can just write "finite Galois extension" wherever Fraleigh wrote "finite normal extension."