Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$.
My problem with this proof is that I do not know how to write it. I understand the idea, but what is the most efficient way to express it?
This was what I came up with: Assume that E is not the splitting field of $f(x)$ over $K$. Then that contradicts by the definition of splitting field.
Is that right conceptually, or do I not understand this? Thanks for any comments or help
Suppose $E$ is not the smallest field containing $K$ and the roots of $f$. Then there is such a field and it is smaller than $E$, say $M$. Now $M$ is a field containing $F$ and the roots of $f$ and is $\subsetneq E$ ...