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).
 A: 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!
A: *

*Any field of positive characteristic. For example, $\mathbb{F}_2 = \{0,1\}$, or more generally $\mathbb{F}_p$ for prime $p$ (integers modulo $p$ with modular addition and multiplication); $\mathbb{F}_{p^n}$, the Galois Field with $p^n$ elements (the splitting field over $\mathbb{F}_p$ of the polynomial $x^{p^n}-x$). Since they have finite characteristic, they cannot be isomorphic to any subfield of $\mathbb{C}$.  There are also infinite fields of positive characteristic, such as $\mathbb{F}_p(x)$, the field of rational functions with coefficients in $\mathbb{F}_p$. 

*Let $X$ be a set of indeterminates of cardinality greater than $\mathfrak{c}$, the cardinality of the continuum. Then $\mathbb{C}(X)$, the field of rational functions on $X$ with coefficients in $\mathbb{C}$ are a field of cardinality $\mathfrak{c}|X| =|X|\gt\mathfrak{c}=|\mathbb{C}|$, so it cannot be isomorphic to a subfield of $\mathbb{C}$. 
A: The field of p-adic numbers is not canonically isomorphic to a subset of $\mathbb C$.
In fact, the claim that the field of p-adic numbers is isomorphic to a subset of $\mathbb C$ is equivalent to some form of choice, therefore not provable in ZF, so p-adic numbers and their completions might also qualify as an example. See e.g. http://en.wikipedia.org/wiki/P-adic_number#Properties
A: $\mathbb Z/p\mathbb Z$ is a (finite) field for every prime $p$. It is not isomorphic to a subfield of $\mathbb C$ since it has characteristic $p$, i.e. in $\mathbb Z/p\mathbb Z$ we have $1+\ldots+1 = 0$. In $\mathbb C$, however, such an equation is not true.
