<s>[INST] What is the minimal polynomial of $\sqrt{3} + \sqrt[3] {2}$ over $\mathbb{Q}$? What is the minimal polynomial of $\sqrt{3} + \sqrt[3]{2}$ over  $\mathbb{Q}$?
I know the basic idea of what a minimal polynomial is--it is the lowest degree monic polynomial in $\mathbb{Q}[x]$ that has the above as a root. But how do you go about calculating it? How do you know intuitively if it does or doesn't exist?
 [/INST] Start with $x=\sqrt{3}+\sqrt[3]{2}$. Then
\begin{gather*}
  x-\sqrt{3} = \sqrt[3]{2} \\
  (x-\sqrt{3})^3 = 2 \\
  x^3 - 3\sqrt{3}x^2 + 9x - 3\sqrt{3} = 2 \\
  x^3 + 9x - 2 = 3\sqrt{3}(x^2+1) \\
  (x^3+9x-2)^2 = 27(x^2+1)^2 \\
  x^6 + 18x^4 - 4x^3 + 81x^2 - 36x + 4 = 27x^4 + 54x^2 + 27 \\
  x^6 - 9x^4 - 4x^3 + 27x^2 - 36x-23 = 0.
\end{gather*}
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