I have the next system of equations where $a_{ij}\in\mathbb{K}$ for a field $\mathbb{K}$ and are not all zero at the same time. Under what circumstances about the field can I ensure that the next system has a nontrivial solution? I can more or less figure that under the reals it should have and, then also, under the complex numbers. But I do not know about other fields and also about the properties I have to ask for this field to get a nonzero solution. I am also not sure about the use of the Hilbert Nullstellensatz in this situation because it affirms that V(I) is not empty... but I know it because $0\in V(I)$ obviously.
\begin{equation}\label{E} \left\{ \begin{aligned} x_{1} &= a_{11}x_{1}^{2}+\cdots+a_{n1}x_{n}^{2}\\ \vdots\\ x_{n} &= a_{1n}x_{1}^{2}+\cdots+a_{nn}x_{n}^{2}. \end{aligned} \right. \end{equation}
What happen in the case $a_{ii}\neq0$ for all $i\in\{1,\dots, n\}$?