[X4]

@omra, why downvote a question? It wasn't a proposed solution to the problem, it was a question on if there is a solution similar to that and otherwise why this would be wrong. I have added that I believe that this would be slow, if you read it carefully.

Btw. "axiom" was probably an unlucky translation of "Körperaxiom". See: http://de.wikipedia.org/wiki/K%C3%B6rper_%28Algebra%29 The english wikipedia says:

"The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication"

[scott_s]

omra likely did not downvote your question.