Difference between revisions of "Dickson invariant"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.E. Dickson, "Linear groups" , Teubner (1901) {{MR|1505871}} {{MR|1500573}} {{ZBL|32.0134.03}} {{ZBL|32.0131.03}} {{ZBL|32.0131.01}} {{ZBL|32.0128.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) {{MR|2333539}} {{MR|2327161}} {{MR|2325344}} {{MR|2284892}} {{MR|2272929}} {{MR|0928386}} {{MR|0896478}} {{MR|0782297}} {{MR|0782296}} {{MR|0722608}} {{MR|0682756}} {{MR|0643362}} {{MR|0647314}} {{MR|0610795}} {{MR|0583191}} {{MR|0354207}} {{MR|0360549}} {{MR|0237342}} {{MR|0205211}} {{MR|0205210}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) {{MR|}} {{ZBL|0221.20056}} </TD></TR></table> |
Revision as of 17:32, 31 March 2012
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
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for ,
(cf. Witt decomposition). Let
![]() |
for any vectors and
from
, and let, for each
,
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Then the following element from :
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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.
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![]() |
then
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References
[1] | L.E. Dickson, "Linear groups" , Teubner (1901) MR1505871 MR1500573 Zbl 32.0134.03 Zbl 32.0131.03 Zbl 32.0131.01 Zbl 32.0128.01 |
[2] | N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210 |
[3] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) Zbl 0221.20056 |
Dickson invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dickson_invariant&oldid=24063