Vol. 15, No. 2, 1965

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On exceptional Jordan division algebras

A. A. Albert

Vol. 15 (1965), No. 2, 377–404

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 = e1FeZF, of two copies eiF of F.

(V) The properties [g(xy)T] = [g(xT)](yT) and xT3 = g1xg, 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 paper1 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 kinds2 of fields F. Thus the case where G is a field, and C may possibly be a division algebra, remained.

Mathematical Subject Classification
Primary: 17.40
Received: 7 April 1964
Published: 1 June 1965
A. A. Albert