# What's the name of this set

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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## 1 Answer

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