Finding the minimal polynomial of $2\sqrt[3]{3}+\sqrt[3]{4}$ in the most efficient way .

Question

How could one find the minimum polynomial of $2\sqrt[3]{3}+\sqrt[3]{4}$, withot using the method where we let $\alpha=2\sqrt[3]{3}+\sqrt[3]{4}$, and keep cubing until we get a polynomial for $\alpha.$

I know that method will work eventually, but the degree of the polynomial is 9 as the degree of the extension is 9 and so it takes far too long.

Any help would be greatly appreciated.

Answer 1

Let $$x:= 2\sqrt[3]{3}+\sqrt[3]{4} = \sqrt[3]{24}+\sqrt[3]{4}$$

Then $$x^3 = 24+4+3(\sqrt[3]{24}\cdot\sqrt[3]{4})(\underbrace{\sqrt[3]{24}+\sqrt[3]{4}}_{=x})= 28+6x\sqrt[3]{12}$$

So $$216\cdot 12x^3 = (x^3-28)^3 =...$$