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.