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.
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