Exercise 3, § 19

Let $(F, s)$ be a semisimple (associative) Cayley K-algebra.

a) Prove that the K-algebra F is simple or isomorphic to the product of a simple algebra and its opposite algebra (consider the action of conjugation on the set of central idempotents).

b) Prove that if F is not commutative, then it is isomorphic to a quaternion algebra over K (use Exercise 1).

c) Prove that if F is commutative, then we are in one of the following cases:

$\alpha ) F = K$, and $s$ is the identity.

$\beta ) F = K\times K$, and $s(\lambda , \mu) = (\mu, \lambda )$.

$\gamma ) F$ is a separable quadratic extension of K, and $s$ is the nontrivial automorphism of F.

$\delta ) K$ has characteristic 2, F is an 2-radical extension of K of height 1, and $s$ is the identity.