Complex Field
Abstract Algebra
For those that remember their Group and Ring/Field Theory, the complex plane is actually a Field with the operations (+,*) defined as within the Complex Number page of this chapter.
Here are some propositions that demonstrate the properties of a Field within the context of complex numbers:
for all 
1) 
2) 
3) 
4) 
5) 
6) 
7) 
8) 
9) 
10) 
11) 
So we have an abelian group with respect to (+) with unit element (0,0) and another abelian group with respect to operation (*) with unit element (1,0)