Let $K$ be a field, and suppose $F$ is a splitting field over $K$ of a (possibly infinite) set of polynomials in $K[x]$. What I want to show is, that if $[F:K]$ is finite, then $F$ is a splitting field over $K$ of a "finite" subset of $K[x]$. This seems intuitively true, but I how do I have to show this?
Finite dimensional splitting field is generated by finite number of polynomials
abstract-algebraextension-fieldfield-theorysplitting-field
Best Answer
Apply the Primite Element Theorem:
Since $F/K$ is Galois and finite, there exists some $\alpha \in F$ such that $F=K(\alpha)$.
Because the extension is algebraic, $F = K[\alpha]$ and $F$ is the splitting field of the minimal polynomial of $\alpha$.
I hope this is what you were going for by requiring the subset of $K[x]$ to be finite in a sense.