Discussion
What is the minimal polynomial of \(\sqrt{2} + \sqrt{3}\) over Q ?
Answer: C
The minimal polynomial is \(x^{4} - 10x^{2} + 1\).
Let \(x = \sqrt{2} + \sqrt{3}\) and square twice:
\[ \begin{aligned} x^{2} &= 5 + 2\sqrt{6} \\ (x^{2} - 5)^{2} &= 24 \\ x^{4} - 10x^{2} + 1 &= 0 \end{aligned} \]
It is minimal because \(\mathbf{Q}(\sqrt{2} + \sqrt{3}) = \mathbf{Q}(\sqrt{2}, \sqrt{3})\) has degree 4 over Q, so the minimal polynomial has degree 4.
That rules out the cubic and the quadratic. \(x^{4} - 6x^{2} + 1\) has roots \(\pm\sqrt{2} \pm 1\), the conjugates of \(1 + \sqrt{2}\), not of \(\sqrt{2} + \sqrt{3}\).
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