## Express some elements in a given field extension

Show that by explicitly writing them as polynomials in .

First, we will find a polynomial satisfied by . Note that the minimal polynomial of over is , and that the minimal polynomial of is . So these elements have conjugates and and , , and . Then is a root of , where . (I recommend using a computer to carry out this multiplication; for instance, WolframAlpha can do it.) We don’t claim that is the minimal polynomial of (it is); all we need to know at the moment is that we only need to consider powers of up to the fifth.

Let , so that . Then , , , and . Now setting and comparing coefficients, we have the following system of linear equations.

Solving this system for appropriate , we have

and

.

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