Q:

Definition of normal field extension

An algebraic field extension $K \subset E$ is called normal if $E$ is the splitting field of a collection of polynomials with coefficients in $K$.
So if $K \subset E$ is some field extension and $E$ is the splitting field of a polynomial $f(x) \in K[x]$, then this extension is normal, because i can take my family of polynomials to be this single polynomial $f(x)$.
Now let $K \subset E$ be a proper but otherwise arbitrary field extension. Let $\alpha \in K$. Then $E$ is the splitting field of $x - \alpha$ and so $K \subset E$ is normal. But this can not be true because $K \subset E$ was arbitrary.
Where is the flaw in the above line of thought?
What am i missing in the definition of a normal extension?
Thanks.

A:

$E$ is not the splitting field of $X-\alpha$. You are misunderstanding the definition of a splitting field. A splitting field of a polynomial $f(X)\in K[X]$ is NOT just a field where your polynomial $f(X)$ splits. It's a field extension $E/K$ s.t. $E$ is generated by roots of $f(X)$. In this case $K[\alpha]=K$, so $E$ is not generated by roots of $X-\alpha$ unless $E=K$.

