Remark 2

Tag 0DNS, permanent. Cite it as [Bourbaki, Tag 0DNS].

Suppose that $A$ is of characteristic $\neq 2$ , and let $(x_1, \ldots, x_m)$ ( $m = 2r$ ) be an orthogonal basis of $E$ . Put

z = 2^r x_1 \ldots x_m \in C(Q);

since

x_i x_j + x_j x_i = 0

for

i \neq j

, we have

zx_j = -x_j z

, which entails that z belongs to the center

Z

C^+(Q)

without belonging to

A

. We have

z^2 = 2^{2r}(-1)^r Q(x_1) \ldots Q(x_m) = (-1)^r D

where D denotes the discriminant of

\Phi

with respect to the basis

(x_j)

(cf. exerc. 9).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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