The Field with four elements
The affine plane
We define the affine space over a ground field to be the set of all tuples with elements in . In the case of we call affine plane. A variety for a subset of polynomials is the set points in which are zeros for each polynomial in . We call a hypersurface, if contains only one element.
We can define a topology on by defining the open sets to be the sets for each . I will not go into detail as to why this results in a topology. As an example, we can always construct

This image shows how solution sets can be united by multiplying the underlying functions. We use the affine real line for this illustration. The green function is the product of the blue and red one and as can be seen, the green graph has exactly the same poles as the red and blue graph combined. Furthermore, we define a hypersurface to be irreducible if the underlying polynomial is irreducible. A hypersurface is reducible if it is not irreducible.
Example of a reducible hypersurface
Consider which is reducible in as it can be factored into the non-units: and .
This can be seen in the graph of , because can be split into two hypersurfaces and .

Example of an irreducible hypersurface
Consider . This defines the circle .

Example of an irreducible hypersurface that is an elliptic curve
Another irreducible hypersurface is in .

This also happens to define an elliptic curve. Also, note that an irreducible hypersurface need not be a connected space in the sense of topology.