12. When a non-empty set of numbers $F$ satisfies the condition "if $a, b \in F$, then $a+b, a-b, ab \in F$, and when $b \neq 0$, $\frac{a}{b} \in F$", $F$ is called a number field. Among the following four propositions about number fields:
(1) 0 is an element of any number field;
(2) If a number field $F$ has a non-zero element, then $2016 \in F$;
(3) The set $P=\{x \mid x=3k, k \in \mathbf{Z}\}$ is a number field;
(4) The set of rational numbers is a number field.
Among them, the true propositions are $\qquad$ (write down the numbers of all true propositions).
12. (1)(2)(4).

According to the definition of a number field, propositions (1), (2), and (4) are all correct.

Take \( a=3, b=6 \), then \( a, b \in P \), but \( \frac{a}{b}=\frac{1}{2} \notin P \), i.e., proposition (3) is incorrect.