Inspired by the construction of the non-mesurable Vitali set I thought about the group $\mathbb{R}/\mathbb{Q}$ (the additive group of the real numbers modulo the rationals). There must be some literature for this, but I haven't found any and it is hard to google. Does anyone know where to find results about this group?
Let $G = \mathbb{R} / \mathbb{Q}$. Is this an interesting group to study?  Is it isomorphic to any more natural mathematical objects?
Can you prove that if $p$ is a prime greater than $5$, then the sum of the squares of the quadratic nonresidues modulo $p$ is divisible by $p$?  Note that I have just proved that the sum of the quadratic residues modulo $p$ is divisible by $p$ for $p$ greater than $3$.