ON THE SPINOR NORM ON UNITARYGROUPS

Abstract

Let
F
be a field of odd characteristic,
E
be a finite extension of
F
equipped an involution with subfield of fixed points
E
0
containing
F
and
V
be a finite dimensional
E
-vector space with a non-degenerate hermi-
tian form
h
. We show a link between the spinor norm in the unitary
group U(
V,h
) and the calculus of determinants and discriminants. Then
we show a formula which links the spinor norm in U(
V,h
) and the spinor
norm in the orthogonal group O(
V,b
h
) defined by a non-degenerate sym-
metric bilinear form
b
h
associated to
h