Name of and references for the equivalence relation $x \sim y :\Longleftrightarrow x^2 = y^2$ Playing around with the concepts of negativity and positivity, I came across the following equivalence relation defined for all elements $x,y$ of a field $\mathbb{F}$:
$$
x \sim y :\Longleftrightarrow x^2 = y^2.
$$
I would like to read more about it; but without a name, I cannot find appropriate references. A superficial search resulted in the concept of square classes. However, this is not really what I was looking for.
Besides, when $\mathbb{F} := \mathbb{R}$, is it correct that the set of non-negative real numbers, $\mathbb{R}_{\geq 0}$, can be considered as the quotient set $(\mathbb{R} / \sim)$?

There is not much to say about this relation and thus it does not deserve a special name:
If $\mathbb F$ has characteristic $2$, it just the equality-relation, since $x^2=y^2 \Longleftrightarrow x=y$.
Otherwise we have $x^2=y^2 \Longleftrightarrow x=y \text{ or } x=-y$, hence the relation pairs any element of the field with its additive inverse.
