Example field of characteristic zero Can you give an example of a field of characteristic zero (other than the complex numbers, real numbers and rational numbers)?
 A: Rational functions with real coefficients form a field of zero characteristic with usual addition and multiplication:
$$\left\{\frac{a_0 + a_1 x + \dots + a_n x^n}{b_0 + b_1 x + \dots + b_m x^m}\right\}$$
In this field, $\left(\frac{P(x)}{Q(x)}\right)^{-1} = \frac{Q(x)}{P(x)}$
A: For any irreducible polynomial $p(x)$ over $\Bbb Q$ (of degree $> 1$), the field $\Bbb Q(\alpha) \cong \Bbb Q[x] / \langle p(x) \rangle$ (where $\alpha$ is a root of $p$) has characteristic $0$ and finite degree $> 1$ over $\Bbb Q$, so it does not coincide with any of $\Bbb Q, \Bbb R, \Bbb C$.
For example, taking $p(x) := x^2 + 1$ gives the field of Gaussian rational numbers, $\Bbb Q[i]$.
A: You can consider "A" like the field of the algebraic numbers.
A: $\mathbb{Q}_p$, the completion of $\mathbb{Q}$ with respect to the $p$-adic absolute value for some prime $p \in \mathbb{Z}$, is a field of characteristic $0$ (it contains $\mathbb{Q}$ as a dense subset).  Interestingly (perhaps), the residue field $\mathbb{Z}_p/p\cdot \mathbb{Z}_p \cong \mathbb{F}_p$ has characteristic $p$ and we say $\mathbb{Q}_p$ has mixed characteristic $(0,p)$.
