## Decide whether a given subset in a field extension is linearly independent

Let $\omega = \frac{-1}{2} + \frac{\sqrt{-3}}{2}$. Decide whether or not each of the following sets is linearly independent over $\mathbb{Q}$: (1) $S_1 = \{\omega, \sqrt{-3}\}$, (2) $S_2 = \{\omega, \omega^2, \omega\sqrt{-3}\}$, and (3) $S_3 = \{2+i,1-3i,12+i\}$.

Suppose we have $a,b \in \mathbb{Q}$ not both zero such that $a\omega + b\sqrt{-3} = 0$. Then $-a/2 + (a/2 + b) \sqrt{-3} = 0$, so that $\sqrt{-3} = a/(a+2b)$. But then $\sqrt{-3}$ is real, a contradiction. So $a = b = 0$, and thus $S_1$ is linearly independent over $\mathbb{Q}$.

Evidently, $\omega - 2\omega^2 + \omega\sqrt{-3} = 0$. (WolframAlpha agrees.) Thus this set $S_2$ is not linearly independent over $\mathbb{Q}$.

Evidently, $37(2+i) - 10(1-3i) + 7(12+i) = 0$. (WolframAlpha agrees.) Thus $S_3$ is not linearly independent over $\mathbb{Q}$.