Prove that the conjugates of an element over a given extension field are distinct

Let \theta = i + \sqrt{2}). Prove that if e \in \mathbb{Q}, then the conjugates of 2 - ei + 2\sqrt{2}i over \mathbb{Q}(\theta) are distinct.


We know from previous exercises that \mathbb{Q}(\theta) has degree 4 over \mathbb{Q}, and that the conjugates of i + \sqrt{2} are \pm i \pm \sqrt{2}.

Using some linear algebra, we see that 2 - ei + 2\sqrt{2}i = 1 + \frac{-e}{6}\theta + \theta^2 + \frac{-e}{6}\theta^3; let p(x) = 1 - x/6 + x^2 - x^3/6. Evaluating p(x) at each of the conjugates of \theta, we see that the conjugates of 2 - ei + 2\sqrt{2}i over \mathbb{Q}(\theta) are 2 - ei + 2\sqrt{2}i, 2 - ei - 2\sqrt{2}i, 2 + ei - 2\sqrt{2}i, and 2 + ei + 2\sqrt{2}i. Provided e \neq 0, these are all distinct.

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