ARITHMETICS OF JORDAN ALGEBRAS 19

be a field such that there exists a central division algebra B of degree 3

over Q and let K be the field of formal power series in T T over £1 . The

reduced norm of B is a homogeneous cubic polynomial f(x x . . . , x ), x,

X

L\

7 1

indeterminates and it is anisotropic. We wish to show that the reduced norm

of G = B SL K is anisotropic and does not represent TT . Let £ , . . . , | e K,

not all zero. Multiplying by a non-zero A e K we may assum e A(°. e o and

that at leas t one 7\£. is a unit of o. Let

u

. be the coefficient of T T in

l " i

A|.; so at leas t one JJ.. + 0. Therefore f(JJ , . . . , |j, ) ± 0 and

A 3 f ( |

1

, |

9

) = f ( A e

r

. . . , A e

9

) * 0. Clearly | A| 3 | f(iy . . ., ^)\ =

I f(A|, , . . . , A|

q

) | = 1 and the reduced norm of G does not represent TT .

It should be noted that exceptional Jordan division algebras do not exist over

local fields. Since homogeneous cubic polynomials in more than 9 variables

are isotropic ([25]) the generic norm of such an algebra, being a homogeneous

cubic polynomial in 27 variables , must be isotropic and # cannot be a

division algebra.