I am studing Galois Theory. I am using the books by Kaplansky, Fields and Rings. I am stuck doing this exercise:
Let $M$ be a split closure of $L$ over $K$ ($M,L,K$ are all fields). Prove that $M=L_1\cup \dots \cup L_r$ (the field generated by the set union, not the set union itself) where $L_i$ is isomorphic to $L$ over $K$.
The actuall problem is that I do not have fully understood the concept of split closure; in the book it is defined in this way.
Let $K \subset L$ be fields and $[L:K]$ finite.There exists a field $M$ containing $L$ such that $M$ is a splitting field over $K$ and no field othen than $M$ between $M$ and $L$ is a splitting field over $K$. If $M_0$ is a second such field, then there exists an isomorphism of $M$ onto $M_0$ which is the identity on $L$. If $L$ is separable then $M$ is normal over $K$.
We shall call a field having the properties of $M$ a split closure of $L$ over $K$. If $L$ is separable we call $M$ normale closure.
Then problem is that the professor in class did not do man examples, so could you please give me some example of split/normal closure, emphasize their difference and the idea behind the introduction of this concept.