## Linear independance over field extensions

Given are n vectors linearly independent over Q.
Furthermore we know they are linearly independent over the field generated by the m-th root of 2.

Now it is intuitively obvious that those n vectors are linearly independent over Q(m-th root of 2).

Is there a theorem stating this fact in a more generalized way or do we have a simple way of proving this?

Thanks,
Jabon