Let be a subfield of , and assume that is separable. We say that a linear algebraic group is defined over when its defining polynomial equations can be chosen with coefficients in . We can form the group of -rational points of . In other words, is the set of fixed points in for the standard action of the Galois group on . The group is called an -form of . More generally, one defines -forms of by ``twisting'' the standard action of the Galois group on by automorphisms of .
Plain examples of -forms of reductive groups are , , the symplectic groups , and the orthogonal groups . Depending on the properties of the field , there may exist many other forms of such ``classical'' matrix groups. For instance, and are real forms of the simple groups and respectively.
In addition, there are certain exceptional simple groups that occur over any field .
A local field is a field equipped with a locally compact non-discrete Hausdorff topology. The local fields of characteristic are , , and finite extensions of the p-adic field . The finite extensions of are called p-adic fields (of characteristic ). Given a local field , a form of -rational points of an algebraic group can be given the topology induced by the topology of . This gives the structure of a locally compact group. There is an important result saying that forms of reductive algebraic groups over local fields are always of type I. This statement provides a large class of examples of type I groups for which the questions raised in the previous section are meaningful.