The field F is usually taken to be the complex numbers C.

A real closed field is a field F in which any of the following equivalent conditions are true:

We have therefore the following invariants defining the nature of a real closed field F:

The next simplest example is the field F itself.

Every field F has some extension which is algebraically closed.

That subgroup can be represented as a linear fractional group on the field F of 11 elements.

The case of K a finite field F is also simple to understand.

The dependence on the field F is only through its characteristic.

First we choose a finite field F with order of at least n, but usually a power of 2.

The notion of a reductive dual pair makes sense over any field F, which we assume to be fixed throughout.