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Dickson invariant

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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=18238
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article