Exercise 7, § 19

Suppose that the characteristic of K is different from 2.

a) Let D be a field containing K and F the quaternion algebra over K of type $(\alpha ,0, \gamma )$. Prove that $D\otimes_KF$ is not a field if and only if there exist two commuting elements $x$ and $y$ in D such that $x^2-\alpha y^2=\gamma$ (apply Exercise 4, c) of VIII, p. 351, first to $D\otimes_KK(i)$ and then to $D\otimes_KF)$.

b) Let $F'$ be another quaternion algebra over K. Then $F\otimes_KF'$ is a field if and only if the only solution of $N_F(q) = N_{F'}(q')$, where $q\in F$ and $q'\in F'$ are pure quaternions (III, §2, p. 618, Exercise 3), is $q=q'= 0$. (Reduce to the case when F is a field and use the criterion of a). Note that if neither $x$ nor $y$ belongs to K, then $y$ belongs to $K(x)$, and write $x=s+q$, where $x\in K$ and $q$ is a pure quaternion.)