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In 1957 the author gave a
construction of a class, of central simple exceptional Jordan algebras H, over any
field F of characteristic not two, called cyclic Jordan algebras. The principal
ingredients of this construction were the following:
(I) A cyclic cubic field K with generating automorphism S over F.
(II) A Cayley algebra C, with K as center, so that C has dimension eight over K,
and dimension 24 over F.
(III) A nonsingular linear transformation T over F of C, which induces S in K,
and commutes with the conjugate operation of C.
(IV) An element g in C, and a nonzero element γ of K, such that g = gT and
gg = [γ(γS)(γS2)]−1. Thus g is nonsingular. Also the polynomial algebra G = F[g] is
either a quadratic field over F or is the direct sum, G = e1F⊕eZF, of two copies eiF
of F.
(V) The properties [g(xy)T] = [g(xT)](yT) and xT3 = g−1xg, for every x and y of
C.
In the present paper we shall give a general solution of the equations of (V), and
shall determine T in terms of two parameters in L = K[g] satisfying some conditions
of an arithmetic type. We shall also provide a special set of values of all of the
parameters of our construction, and shall so provide a proof of the existence of
cyclic Jordan division algebras with attached Cayley algebra C a division
algebra.
The existence of a transformation T with the two properties of (V) for some element
g = gT, in the Cayley algebra C which satisfies (IV), was demonstrated by the author in the
1957 paper
only in the case where G is not a field, and consequently C is a split algebra. In that
case it was proved that cyclic Jordan division algebras do exist, for certain
kinds
of fields F. Thus the case where G is a field, and C may possibly be a division
algebra, remained.
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