In
mathematics, a
reductive group is an
algebraic group G over an algebraically closed field such that the
unipotent radical of
G is trivial (i.e., the group of unipotent elements of the
radical of
G). Any
semisimple algebraic group is reductive, as is any
algebraic torus and any
general linear group. More generally, over fields that are not necessarily algebraically closed, a reductive group is a smooth affine algebraic group such that the unipotent radical of
G over the algebraic closure is trivial. The intervention of an algebraic closure in this definition is necessary to include the case of imperfect ground fields, such as local and global function fields over finite fields. Algebraic groups over (possibly imperfect) fields
k such that the
k-unipotent radical is trivial are called
pseudo-reductive groups.