<s>[INST] Minimal polynomial of $\frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$ I am struggling to find the minimal polynomial for $\displaystyle \frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$.
Does anyone have any suggestions?
Thanks,
Katie.
 [/INST] One can compute the minimal polynomial using resultants or Grobner bases. But that is a bit overkill here since it can be done fairly straightforwardly by hand. Namely, let $\rm\ y = \sqrt{2}+\sqrt[3] 5\:.\ $ Then $\rm\: (y-\sqrt 2)^3 = 5\:,\:$ i.e. $\rm\:y^3 + 6\ y - 5 - (3\ y^2 + 2)\ \sqrt{2} = 0\:.\:$ Multiplying that by its conjugate yields $\rm\: y^6 - 6\ y^4 -10\ y^3 + 12\ y^2 -60\ y + 17 = 0\:.\:$ Putting $\rm\ y = \sqrt{3}\ x\:,\:$ then multiplying that by its conjugate yields $\rm\:729\ x^{12} -2916\ x^{10} + 4860\ x^8 - 5670\ x^6 -11340\ x^4 - 9576\ x^2 + 289 = 0\:.$
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