If the field K is not algebraically closed, the theorem can fail.

Assume that R is a field K of characteristic 0.

Over a general field k, there is no good notion of singular (co)homology.

A differential field is a field K, together with a derivation.

Consider the action of S on affine 6-space over the field k with 3 elements.

Let R be an algebra over a field k that is an integral domain.

Let V be a vector space over a field k and fix a basis for it.

It is often written V or when the field k is understood.

The case of a global field K is addressed by the global class field theory.

In mathematics, a ground field is a field K fixed at the beginning of the discussion.