Proposition 3

Let $\alpha ,\beta ,\gamma$ be elements of K, and let $f$ be a quaternion algebra of type $(\alpha , \beta , \gamma )$. Denote its Cayley trace and norm by $T_F$ and $N_F$. The following properties are equivalent:

(i) The algebra F is central simple.

(ii) For every nonzero element $x$ of F, there exists an element $y$ in F such that $T_F(xy)\not= 0$.

(iii) We have $(4\alpha +\beta^2)\gamma \not= 0$.

Suppose that these properties hold. Then for every $x$ in F, the reduced characteristic polynomial of $x$ is $X^2-T_F(x)X + N_F(x)$. In particular, $T_F(x)$ is the reduced trace of $x$, and $N_F(x)$ is its reduced norm.

(i)⇒(ii): If the algebra F is central simple, then it follows from Proposition 1 of VIII, p. 361 and the definition of the reduced trace (VIII, p. 340, Definition 2) that $T_F$ is its reduced trace; assertion (ii) follows (VIII, p. 343, Proposition 5).

(ii)⇔(iii): Let $(e_i)_{1 \leq i \leq 4}$ be a basis of F of type $(\alpha, \beta, \gamma)$ (III, §2, No. 5, p. 445). The matrix $(T_F(e_i e_j))$ is equal to

Its determinant is $-\gamma^2 (4\alpha + \beta^2)^2$. The equivalence of properties (ii) and (iii) follows from V, §8, No. 2, p. 49, Lemma 1.

(iii)⇒(i): Suppose $(4\alpha + \beta^2)\gamma \neq 0$. We then have $\gamma \neq 0$, and we have $\beta \neq 0$ if K has characteristic 2. By Proposition 2, the algebra F is central. Let x be an element of the Jacobson radical of F. For every $y \in F$, $xy$ is nilpotent, so $T_F(xy) = 0$ (Remark 3 of VIII, p. 362). Since (ii) is equivalent to (iii), we have $x = 0$. This proves that F is a semisimple K-algebra. Since its center is K, it is simple.

The last assertion follows from Proposition 1 of VIII, p. 361 and the definition of the reduced characteristic polynomial (VIII, p. 340, Definition 1).

Denote the characteristic of K by p. By Proposition 3, if $p \neq 2$, then every quaternion algebra over K of type $(\alpha, 0, \gamma)$ with $\alpha$ and $\gamma$ in $K^*$ is central simple. If $p = 2$, then every quaternion algebra of type $(\alpha, 1, \gamma)$ with $\alpha \in K$ and $\gamma \in K^*$ is central simple. Conversely, we have the following.