Editing Linear algebraic group (section)

==Semisimple and unipotent elements==
{{main|Jordan–Chevalley decomposition}}
For an algebraically closed field ''k'', a matrix ''g'' in ''GL''(''n'',''k'') is called '''semisimple''' if it is [[diagonalizable]], and '''[[unipotent]]''' if the matrix ''g'' − 1 is [[nilpotent]]. Equivalently, ''g'' is unipotent if all [[eigenvalue]]s of ''g'' are equal to 1. The [[Jordan canonical form]] for matrices implies that every element ''g'' of ''GL''(''n'',''k'') can be written uniquely as a product ''g'' = ''g''<sub>ss</sub>''g''<sub>u</sub> such that ''g''<sub>ss</sub> is semisimple, ''g''<sub>u</sub> is unipotent, and ''g''<sub>''ss''</sub> and ''g''<sub>u</sub> [[commuting matrices|commute]] with each other.

For any field ''k'', an element ''g'' of ''GL''(''n'',''k'') is said to be semisimple if it becomes diagonalizable over the algebraic closure of ''k''. If the field ''k'' is perfect, then the semisimple and unipotent parts of ''g'' also lie in ''GL''(''n'',''k''). Finally, for any linear algebraic group ''G'' ⊂ ''GL''(''n'') over a field ''k'', define a ''k''-point of ''G'' to be semisimple or unipotent if it is semisimple or unipotent in ''GL''(''n'',''k''). (These properties are in fact independent of the choice of a faithful representation of ''G''.) If the field ''k'' is perfect, then the semisimple and unipotent parts of a ''k''-point of ''G'' are automatically in ''G''. That is (the '''Jordan decomposition'''): every element ''g'' of ''G''(''k'') can be written uniquely as a product ''g'' = ''g''<sub>ss</sub>''g''<sub>u</sub> in ''G''(''k'') such that ''g''<sub>ss</sub> is semisimple, ''g''<sub>u</sub> is unipotent, and ''g''<sub>''ss''</sub> and ''g''<sub>u</sub> commute with each other.<ref>Milne (2017), Theorem 9.18.</ref> This reduces the problem of describing the [[conjugacy class]]es in ''G''(''k'') to the semisimple and unipotent cases.