The definition of a normal extension in the book "Abstract algebra" is :
If $K$ is an algebric 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
I think that there is something here I don't understand: If $K$ is an algebraic extension of $F$ then by definition each element of $K$ is a root of a polynimial with coefficients in $F$.
So each element of $K$ corresponds to a polynomial in $F[x]$ (s.t the element is a root of this polynomial).
So I deduced that $K$ is the splitiing field of the collections of polynomials in $F[x]$ that corresponds to the elements in $K$. Hence every algebric extension is also a normal one.
What part of my argument is wrong ?