Maximal subfield of an algebra

Let $A$ be an associative unital algebra over a field $F$ . A maximal subfield of $A$ is a field extension $L/F$ realized as a subalgebra $L\subseteq A$ with the same identity, such that $L$ is not properly contained in another field subalgebra of $A$ containing $F1_A$ .

Maximality is by inclusion among field subalgebras. It does not mean that $L$ is algebraically closed or is a maximal proper subring of $A$ .

Quaternion division algebras

In a quaternion division algebra over a field of characteristic different from two, each nonscalar element $x$ generates a quadratic field $F[x]$ , and these are exactly the maximal subfields. The reduced characteristic polynomial has degree two, and division excludes a product decomposition or a nonzero nilpotent.

For the rational Hamilton algebra, $\mathbb Q(i)$ and $\mathbb Q(i+j+k)\cong\mathbb Q(\sqrt{-3})$ are examples. The latter follows from $(i+j+k)^2=-3$ .

Split warning

A nonscalar element of $M_2(F)$ need not generate a field. For example $\operatorname{diag}(0,1)$ generates a subalgebra isomorphic to $F\times F$ . The division hypothesis in the preceding characterization is essential.

References

John Voight, Quaternion Algebras, author's text, Corollary 4.4.5 and Proposition 7.7.8, quadratic subfields and centralizers.
The rational examples follow by direct quaternion multiplication; the split example follows by polynomial evaluation at $0$ and $1$ .