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 disser ...

, 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. Man ...

''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 field ...

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 i ...

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 compact t ...

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

* {{DEFAULTSORT:Kneser-Tits conjecture Algebraic groups Conjectures