Editing Linear algebraic group (section)

==Tori==
{{main|Algebraic torus}}
A '''torus''' over an algebraically closed field ''k'' means a group isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup>, the [[Cartesian product|product]] of ''n'' copies of the multiplicative group over ''k'', for some natural number ''n''. For a linear algebraic group ''G'', a '''maximal torus''' in ''G'' means a torus in ''G'' that is not contained in any bigger torus. For example, the group of diagonal matrices in ''GL''(''n'') over ''k'' is a maximal torus in ''GL''(''n''), isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup>. A basic result of the theory is that any two maximal tori in a group ''G'' over an algebraically closed field ''k'' are [[conjugacy class#Conjugacy of subgroups and general subsets|conjugate]] by some element of ''G''(''k'').<ref>Borel (1991), Corollary 11.3.</ref> The '''rank''' of ''G'' means the dimension of any maximal torus.

For an arbitrary field ''k'', a '''torus''' ''T'' over ''k'' means a linear algebraic group over ''k'' whose base change <math>T_{\overline k}</math> to the algebraic closure of ''k'' is isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup> over <math>\overline k</math>, for some natural number ''n''. A '''split torus''' over ''k'' means a group isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup> over ''k'' for some ''n''. An example of a non-split torus over the real numbers '''R''' is
:<math>T=\{(x,y)\in A^2_{\mathbf{R}}: x^2+y^2=1\},</math>
with group structure given by the formula for multiplying complex numbers ''x''+''iy''. Here ''T'' is a torus of dimension 1 over '''R'''. It is not split, because its group of real points ''T''('''R''') is the [[circle group]], which is not isomorphic even as an abstract group to ''G''<sub>''m''</sub>('''R''') = '''R'''*.

Every point of a torus over a field ''k'' is semisimple. Conversely, if ''G'' is a connected linear algebraic group such that every element of <math>G(\overline k)</math> is semisimple, then ''G'' is a torus.<ref>Milne (2017), Corollary 17.25</ref>

For a linear algebraic group ''G'' over a general field ''k'', one cannot expect all maximal tori in ''G'' over ''k'' to be conjugate by elements of ''G''(''k''). For example, both the multiplicative group ''G''<sub>''m''</sub> and the circle group ''T'' above occur as maximal tori in ''SL''(2) over '''R'''. However, it is always true that any two '''maximal split tori''' in ''G'' over ''k'' (meaning split tori in ''G'' that are not contained in a bigger ''split'' torus) are conjugate by some element of ''G''(''k'').<ref>Springer (1998), Theorem 15.2.6.</ref> As a result, it makes sense to define the ''' ''k''-rank''' or '''split rank''' of a group ''G'' over ''k'' as the dimension of any maximal split torus in ''G'' over ''k''.

For any maximal torus ''T'' in a linear algebraic group ''G'' over a field ''k'', Grothendieck showed that <math>T_{\overline k}</math> is a maximal torus in <math>G_{\overline k}</math>.<ref>Borel (1991), 18.2(i).</ref> It follows that any two maximal tori in ''G'' over a field ''k'' have the same dimension, although they need not be isomorphic.