Definition of normal extension on Abstract Algebra of Dummit and Foote:
Definition. If $K$ is an algebraic extension of $F$ which is the splitting field over $F$ for a collection of polynomials $f(x)\in F[x]$ then $K$ is called a normal extension of $F$.
We shall generally use the term ''splitting field'' rather than normal extension of $F$.
Is ''collection'' the difference between normal extension and splitting field ?
If there are finite polynomials in the collection we can just identify the extension as the splitting field of the product of polynomials in the collection, if there are infinite polynomials we cannot identify the extension as a splitting field and that's the reason of using the term ''collection''.
Is my understanding right? If so, why can we substitute ''splitting field'' for ''normal extension'' ?