I'm looking to come up with a $\mathcal{L}_{ring}{-sentence}$ which have constant symbols $0, 1$ and binary functions $+, \times$.
I'd like to find a sentence using this structure that compares $\mathbb{Z}$ and $\mathbb{C}$ such that it satisfies one but not the other.
I understand the differences, I'm just having a hard time expressing it in terms of logic..
To give an example using $(\mathbb{Z}, \mathbb{Q})$ we can define $\phi = \forall x \forall y\exists z(y\times z =x).$ This statement only satisfies the rationals, but does not necessarily for integers.
How do I come up with something along these lines for the complex numbers?
Would something like
$\forall x ( x\times x=(0-1))$ work?
The only issue is that I don't have a subtraction operator.
$\exists x ( x\times x + 1 = 0)$
In $\mathbb{C}$ the formula $\exists x: x \times x + 1 = 0$ holds, which is expressed in terms of $0,1, +, \times$. This does not hold in $\mathbb{Z}$, nor in $\mathbb{Q}$ or $\mathbb{R}$.
But maybe easier : $\forall x : \lnot (x = 0) \to (\exists y: x \times y =1)$ does not hold in $\mathbb{Z}$ (as it is not a field), but it does hold in $\mathbb{R}, \mathbb{C}, \mathbb{Q}$ or any other field.