<s>[INST] Is $\left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$ an integer? The problem is the following:

Prove that this number
  $$x = \left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$$
  is an integer. Show which integer it is.

I  thought that it has some relations with something like complex numbers such as the set $N(\sqrt{2})$ or the some kind of integer polynomial which has root $x$ and therefore show that it has integer solutions.
 [/INST] Let $a,b$ be the first and second term of the sum then $x = a+b$. Observe that: $ab = 7 \to x^3 = (a+b)^3 = a^3+b^3 + 3ab(a+b)=90+3\times 7\times x \to x^3-21x-90 = 0\to (x-6)(x^2+6x+15)=0\to (x-6)((x+3)^2+6)=0\to x = 6.$
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