group over an arbitrary field k is called semisimple or reductive if the base change is semisimple or reductive, where is an algebraic closure of k. (This is equivalent to the definition of reductive groups in the introduction when k is perfect.) Any torus over k, such as the multiplicative group Gm, is reductive. With representation theory Over fields of characteristic zero another equivalent definition of a reductive group is a connected group admitting a faithful semisimple representation which remains semisimple over its algebraic closure page 424. Simple reductive groups A linear algebraic group G over a field k is called simple (or k-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup of G over k is trivial or equal to G. (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer n at least 2 and any field k, the group SL(n) over k is simple, and its center is the group scheme μn of nth roots of unity. A central isogeny of reductive groups is a surjective homomorphism with kernel a finite central subgroup scheme. Every reductive group over a field admits a central isogeny from the product of a torus and some simple groups. For example, over any field k, It is slightly awkward that the definition of a reductive group over a field involves passage to the algebraic closure. For a perfect field k, that can be avoided: a linear algebraic group G over k is reductive if and only if every smooth connected unipotent normal k-subgroup of G is trivial. For an arbitrary field, the latter property defines a pseudo-reductive group, which is somewhat more general. Split-reductive groups A reductive group G over a field k is called split if it contains a split maximal torus T over k (that is, a split torus in G whose base change to is a maximal torus in ). It is equivalent to say that T is a split torus in G that is maximal among all k-tori in G. These kinds of groups are useful because their classification can be described through combinatorical data called root data. Examples GLn and SLn A fundamental example of a reductive group is the general linear group of invertible n ×