Let's take field $(K, +,\times,0,1)$. Let's take set $S(K) \subset K$ such that:
\[ \forall_{e \in K}\exists_{a \in S(K)}\exists_{p \in K} e = a \times p \times p \] \[ \lnot \exists_{a, b \in S(K)}\exists_{p \in K} (a \times p \times p = b \land a \neq b) \]
For example $S(\mathbb{R}) = \{-1, 1\}$, $S(\mathbb{C}) = \{1\}$, $S(\mathbb{Z}_3) = \{1, 2\}$, $S(\mathbb{Z}_5) = \{1, 2\}$. Of course sets are not unique like $S(\mathbb{R}) = \{-e,\pi\}$ is also possible.
(Sorry if I misuse terms - I haven't found english translation - but I found it useful to generalize the signature of quadratic polynomials).
