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.

Let $u=\sqrt[3]{45+29\sqrt2}$ and $v=\sqrt[3]{45-29\sqrt2}$; so, $x=u+v$. We have
$$
u^3 + v^3=90,\\
uv = \sqrt[3]{45+29\sqrt2}\cdot \sqrt[3]{45-29\sqrt2} = \sqrt[3]{45^2-29^2\cdot2} = \sqrt[3]{343} = 7
$$
But
$$
u^3 + v^3 = (u+v)(u^2 - uv + v^2) = (u+v)((u+v)^2 - 3uv),
$$
and
$$
90 = x(x^2 - 21).
$$
Since $x$ must be integer, check divisors of $90$. $x^2 - 21 > 0$, so $x \ge 5$. Try $x=6$:
$$
90 = 6(6^2-21) = 6\cdot 15.
$$
