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.