## Minimal polynomial

I am asking a question about Sheet 2 Problem 9 (solutions are provided for up to 8).

Why and are conjugates of ?

I’ve managed to find minimal polynomial using DeMoivre’s Theorem and I’ve shown that , satisfy it (). But the question asks to do the opposite: show that they are conjugates and hence find the minimal polynomial of over .

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In the problem, you have , where Thus, , where we use the bar to denote the complex conjugate. Now, the key point is this: every embedding of into is the restriction of an embedding of into , except some of the embeddings will become equal when restricted. Now, has six embeddings for such that . This can be seen by considering the roots of the minimal polynomial for . Since , we see that for each embedding, . Therefore, the conjugates of are simply . Hence, you get exactly the three numbers indicated.

By the way, the minimal polynomial needs to be monic.

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