Q:

If $M_1:K$ and $M_2:K$ are normal extensions then $K(M_1,M_2):K$ is normal.

By normal extension of $K$ I mean the splitting field for a $S\subseteq K[x]$, and by $K(M_1,M_2)$ I mean the least field that contains $K, M_1,$ and $M_2$. (Suppose $L:M_1:K$ and $L:M_2:K$ so that this makes sense.)
I $\textit{guessing}$ that if $S_1\subseteq K[x]$ and $S_2\subseteq K[x]$ are sets such that $M_1$ and $M_2$ are their splitting fields over $K$ respectively, then $K(M_1,M_2)$ is probably a splitting field for $S_1\cup S_2$. The elements of $S_1\cup S_2$ definitely split over $K(M_1,M_2)$, but I'm having trouble checking that if $K(M_1,M_2):H:K$ and $H\subsetneq K(M_1,M_2)$ then there's some element of $S_1\cup S_2$ that doesn't split over $H$.

A:

Definitions being used: (I state them because apparently my book using non-standard terminology.)

$L:K$ means $L$ extends $K$. (Contains $K$ as a subfield.)
$L:K$ is called normal if there exists some $S\subseteq K[x]$ such that $L$ is the least field extension of $K$ where every element of $S$ splits. (i.e. $L$ is a splitting field for $S$.)
If $L:K$, we say a subset $S\subseteq K[x]$ splits over $L$ if every element of $S$ splits over $L$.

We will write $M_1M_2$ for $K(M_1,M_2)$.
Let $S_1\subseteq K[x]$ and $S_2\subseteq K[x]$ be sets such that $M_1$ and $M_2$ are their splitting fields over $K$ respectively. Then, $S_1\cup S_2$ clearly splits over $M_1M_2$ but let's check minimality. Suppose $H$ is a field such that $H:K$ and $S_1\cup S_2$ splits over $H$. In particular $S_1$ splits over $H$ and so, by minimality of $M_1$, we have $H:M_1$. Analogously, we get $H:M_2$ and combining these we get $H:M_1M_2$.

