A construction used in the study of quadratic forms over fields of characteristic 2, which allows one, in particular, to introduce analogues of the special orthogonal group over such fields. In fact, a Dickson invariant is an element 
of a field 
of characteristic 2 associated to any similarity 
of a countable-dimensional vector space 
over 
with respect to the symmetric bilinear form 
associated with a non-degenerate quadratic form 
on 
. Introduced by L.E. Dickson [1].
By virtue of the condition imposed on the characteristic of the field, the form 
is alternating and there exists a basis 
in 
for which
for 
, 
(cf. Witt decomposition). Let
for any vectors 
and 
from 
, and let, for each 
,
Then the following element from 
:
is called the Dickson invariant of the similarity 
with respect to the basis 
. For 
to be a similarity with respect to 
with similarity coefficient 
(i.e. 
for any vector 
) it is necessary and sufficient that 
or that 
. Similarities 
with respect to 
for which 
are called direct similarities. The direct similarities form a normal subgroup of index 2 in the group of all similarities with respect to 
.
If 
is the form defined by 
for any vector 
, and if 
and 
are the pseudo-discriminants of these forms with respect to the basis 
, i.e.
then
References
| [1] | L.E. Dickson, "Linear groups" , Teubner (1901) |
| [2] | N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) |
| [3] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) |
How to Cite This Entry:
Dickson invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dickson_invariant&oldid=24063
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
