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 $(x,y),(a,b),(c,d)\epsilon \mathbb{C}$ 1) $(x,y)+(a,b)=(x+a,y+b)$ 2) $((x,y)+(a,b))+(c,d)=(x,y)+((a,b)+(c,d))$ 3) $(x,y)+(a,b)=(a,b)+(x,y)$ 4) $(x,y)+(0,0)=(x,y)$ 5) $(x,y)+(-x,-y)=(0,0)$ 6) $(x,y)*((a,b)+(c,d))=(x,y)(a,b)+(x,y)(c,d)$ 7) $(x,y)*(a,b)=(xa-yb,xb+ya)$ 8) $(x,y)((a,b)(c,d))=(x,y)(a,b)(c,d)$ 9) $(x,y)(a,b)=(a,b)(x,y)$ 10) $(x,y)(1,0)=(x,y)$ 11) $(x,y)(x/(x^2+y^2),-y/(x^2+y^2))=(1,0)$ 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)

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