Is any algebraic extension a subfield of a spitting field?

Question

Let $F \subset K$ be an algebraic field extension, is $K$ a subfield of a splitting field of $S=\{{f_i}\}_{i \in I} $ over $F$? I tried: Let $\{{u_i}\}$ be a $F$-basis of K, $\{{p_i}\}$ be its corresponding minimal polynomials ,can we deduce that $K$ is a subfield of the splitting field of $\{{p_i}\}$ over $F$? I got somewhat confused about the underlying sets. Thanks in advance.

Answer 1

Zorn's lemma (see Existence of extension of an embedding for an algebraic extension of a field) implies that there is an embedding $K \rightarrow \overline{F}$.

Now, $\overline{F}$ is the splitting field for $S = \text{every polynomial in $F[x]$}$ , and $K$ is isomorphic to the image of the embedding in $\overline{F}$.