In mathematics, the Kneser–Tits problem, introduced by based on a suggestion by
Martin Kneser
Martin Kneser (21 January 1928 – 16 February 2004) was a German mathematician. His father Hellmuth Kneser and grandfather Adolf Kneser were also mathematicians.
He obtained his PhD in 1950 from Humboldt University of Berlin with the disse ...
, asks whether the Whitehead group ''W''(''G'',''K'') of a semisimple simply connected isotropic
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
...
''G'' over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'' is trivial. The Whitehead group is the quotient of the
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fie ...
s of ''G'' by the
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
generated by ''K''-subgroups isomorphic to the additive group.
Fields for which the Whitehead group vanishes
A special case of the Kneser–Tits problem asks for which fields the Whitehead group of a semisimple almost simple simply connected
isotropic algebraic group is always trivial.
showed that this Whitehead group is trivial for
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ...
s ''K'', and gave examples of fields for which it is not always trivial. For global fields the combined work of several authors shows that this Whitehead group is always trivial .
References
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External links
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{{DEFAULTSORT:Kneser-Tits conjecture
Algebraic groups
Conjectures