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