I know this has been debated before but I just want a clear answer whether i^2=-1 is defined or can be proved by simpler axioms? If the latter than how would you prove it? I am guessing that it is defined and can't be proved. It is defined so that all polynomials such as x^2+1=0 which does not have a root in R has one (or some) in a new field we define as C. Moreoever, in this field we introduce and define a new quantity called i. Which when squared gives -1 and so i^2+1=0.