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}.

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