Find an integer multiple of a given algebraic number which is an algebraic integer

Let \theta = \omega/9 + \sqrt{2}/5, where \omega is a root of (x^7-1)/(x-1). Find a rational integer k such that k\theta is an algebraic integer.

Recall that sums and products of algebraic integers are algebraic integers.

Note that \omega is an algebraic integer, as is \sqrt{2}, and thus 5\omega and 9\sqrt{2} are algebraic integers. So 45\theta = 5\omega + 9\sqrt{2} is an algebraic integer.

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