Just out of curiosity - when we define a field, why bother mention multiplication, when its nothing more then repeating the same addition operation?
Here's the definition we were taught in calculus for physicists class:
A field F is a non-empty set on which two binary operations are defined: an operation which we call addition, and denote by +, and an operation which we call multiplication and denote by $\cdot$ (or by nothing, as in a b = ab). The operations on elements of a field satisfy nine defining properties, which we list now...
And then of course you have the axioms of field.