TITLE: What is the first order axiom characterizing a field having characteristic zero?
QUESTION [6 upvotes]: In this thread on the axioms of $\mathbb Q$ it's stated that a field having characteristic zero can be written down in first-order logic. 
The definition in the logic lecture notes I work with (by Stephen G. Simpson) give the standard definition, which isn't a formal first order statement:
$$"(1+1+\ \dots\ +1+1)_{n\  \text{times}}\neq0"$$
Now in this Wikipedia page on periodic groups (and in the reference given there, I looked it up), it is stated that a group being periodic 
$$"\forall x.\, ((x=e) \lor (x\circ x=e) \lor ((x\circ x)\circ x=e) \lor \ldots)"$$
can't be stated in first-order logic. 
Both would be "for all $n\in \mathbb N$" statements and it makes my question how the characteristic zero axiom can be a first-order statement and secondly if we can formally pull off $\mathbb Q$ without first formalize $\mathbb N$.
My question is: What is the first order axiom characterizing a field having characteristic zero?

REPLY [6 votes]: The problem with the group theory case is that there is no natural way to break up an infinite $\lor$ into separate axioms. But the infinite axiom scheme for $\mathbb Q$ is representing an infinite $\land$, so we can break it up into separate axioms.