Example of fields that are not subsets of the complex numbers I've read the axioms of a field. To understand the generality of the axioms, could you give me an example of a field which is not (isomorphic to) a subset of complex number (with or without modulus operations).

Let $I$ be a set of cardinality greater than the cardinality of the reals (or equivalently the complex numbers).  For every $i$ in $I$, invent a symbol $x_i$.  Let $F$ consist of all ratios of polynomials in the $x_i$. By polynomial we mean say polynomial with real coefficients. Any polynomial will "mention" only finitely many of the $x_i$. Add and multiply in the natural way.
Remark: The "with or without modulus operations" seems to allow the possibility of taking a subring of the complex numbers, defining an equivalence relation on it ("modulus") and forming the quotient structure. It may be very difficult to describe what sorts of fields can arise in this way. That's why I took the safe route and made my $F$ so big that it could never be produced by this kind of process!
