Editing Linear algebraic group (section)

==Classification of reductive groups==
{{main|Reductive group}}
Reductive groups include the most important linear algebraic groups in practice, such as the [[classical group]]s: ''GL''(''n''), ''SL''(''n''), the [[orthogonal group]]s ''SO''(''n'') and the [[symplectic group]]s ''Sp''(2''n''). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them. Remarkably, [[Claude Chevalley]] gave a complete classification of the reductive groups over an algebraically closed field: they are determined by [[root data]].<ref>Springer (1998), 9.6.2 and 10.1.1.</ref> In particular, simple groups over an algebraically closed field ''k'' are classified (up to quotients by finite central subgroup schemes) by their [[Dynkin diagram]]s. It is striking that this classification is independent of the characteristic of ''k''. For example, the [[exceptional Lie group]]s ''G''<sub>2</sub>, ''F''<sub>4</sub>, ''E''<sub>6</sub>, ''E''<sub>7</sub>, and ''E''<sub>8</sub> can be defined in any characteristic (and even as group schemes over '''Z'''). The [[classification of finite simple groups]] says that most finite simple groups arise as the group of ''k''-points of a simple algebraic group over a finite field ''k'', or as minor variants of that construction.

Every reductive group over a field is the quotient by a finite central subgroup scheme of the product of a torus and some simple groups. For example,
:<math>GL(n)\cong (G_m\times SL(n))/\mu_n.</math>

For an arbitrary field ''k'', a reductive group ''G'' is called '''split''' if it contains a split maximal torus over ''k'' (that is, a split torus in ''G'' which remains maximal over an algebraic closure of ''k''). For example, ''GL''(''n'') is a split reductive group over any field ''k''. Chevalley showed that the classification of ''split'' reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. For example, every nondegenerate [[quadratic form]] ''q'' over a field ''k'' determines a reductive group ''SO''(''q''), and every [[central simple algebra]] ''A'' over ''k'' determines a reductive group ''SL''<sub>1</sub>(''A''). As a result, the problem of classifying reductive groups over ''k'' essentially includes the problem of classifying all quadratic forms over ''k'' or all central simple algebras over ''k''. These problems are easy for ''k'' algebraically closed, and they are understood for some other fields such as [[number field]]s, but for arbitrary fields there are many open questions.