Definition of splitting field: Why do we require centrality?

Question

Let $k$ be a field. Let $D$ be a division algebra over $k$. Call a field extension $K/k$ a splitting field for $D$ if there exists a positive integer $n$ such that $D\otimes_k K\cong M(n\times n,K).$ Most of the definitions of a splitting field additionally require that $D$ is a central $k$-algebra, i.e. $Z(D)=k$. Why is that?

Answer 1

Given a $k$-algebra $A$ (finite dimensional over $k$) and a field extension $K/k$, a theorem says that $A\otimes_k K$ is central simple over $K$ iff and only if $A$ is central simple over $k$. Now $M_n(K)$ is central simple over $K$, so if you believe this theorem, this implies that your $D$ is forced to be central simple, if you want to have any chance for your isomorphism to hold.

The simplicity is guaranteed by the division algebra property.

Centrality is not automatic.

Let me emphasise that centrality is crucial, since otherwise you have easy counterexamples: let $L/k$ be an arbitrary field extension of finite degree >1, and let $D=L$.

Then $D$ has no splitting field. Indeed, $D\otimes_k K$ is commutative, so, if a splitting field exists, $n=1$(the $n\times n$ matrix ring is not commutative if $n\geq 2$). Hence $\dim_K(D\otimes_k K)=1=\dim_k(D)$; this is a contradiction since the degree of $D=L$ over $k$ is $>1$.