In this paper, we show that for each cubic algebraic number \(\alpha \) there is a nonzero \(\gamma \in {{\mathbb {Q}}}(\alpha )\) such that for all \(u, v \in {{\mathbb {Q}}}\) the product \((u+v\alpha )\gamma \) is not a square in the field \({{\mathbb {Q}}}(\alpha )\) except for the trivial case \(u=v=0\) . This completes some earlier investigations in which such a result has been proved for most, but not all, cubic numbers \(\alpha \) . The method is constructive and allows us to find such a number \(\gamma \) explicitly. Our result implies that each cubic field K contains linear hyperplanes H that do not contain elements of the form \(\omega ^2\) , where \(\omega \in K \setminus \{0\}\) . We show that the same is true for all totally real fields K as well. However, for each \(d \ge 5\) there are fields K of degree d such that each linear hyperplane H in K has an infinite intersection with \(K^2=\{ \omega ^2 \>|\> \omega \in K\}.\)
