Exercise 2, § 6

Let $D$ be a non-commutative field of characteristic $\neq 2$ and $\mathbf{Z}$ its centre. Suppose that every commutative subfield $K$ of $D$ containing $\mathbf{Z}$ is of degree $\leq 2$ over $\mathbf{Z}$.

(a) Show without using Lemma 1 that $[D : \mathbf{Z}] = 4$. (Let $a \in D - \mathbf{Z}$ be such that $a^2 \in \mathbf{Z}$ and let $\sigma(x) = axa^{-1}$ for all $x \in D$; note that $D$ decomposes as a direct sum of two vector subspaces over $\mathbf{Z}$, $D_+$ and $D_-$ such that $\sigma(x) = x$ on $D_+$, $\sigma(x) = -x$ on $D_-$; note also that $D_+$ is a subfield of $D$ and $D_-$ is a 1-dimensional vector space over $D_+$; finally, show that $D_+$ cannot be distinct from $\mathbf{Z}(a)$.) (b) Show that $D$ is a quaternion algebra over $\mathbf{Z}$. (Form a basis of $D$ over $\mathbf{Z}$ with the aid of (a).)