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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).

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If I am deciphering your notation correctly, I would call $S$ "a set of coset representatives for $K^{\times}/(K^{\times})^2$."

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  • $\begingroup$ I haven't check it fully but from what I understend it looks like this. $\endgroup$ – Maciej Piechotka Aug 9 '10 at 20:28

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