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**hg305** I'm having trouble showing that a particular number is an algebraic integer in the cubic number field $\displaystyle \mathbb{Q}(\alpha)$ where $\displaystyle \alpha^3=m=hk^2\in\mathbb{Z}$ and $\displaystyle m$ is cube-free and $\displaystyle h$ and $\displaystyle k$ are both square-free. There is also the extra condition that $\displaystyle m\equiv1\mod9$.

I'm trying to show that $\displaystyle \frac{\alpha^2+\alpha k^2+k^2}{3k}$ is an algebraic integer, given that $\displaystyle \frac{\alpha^2}{k}$ and $\displaystyle \frac{(\alpha-1)^2}{3}$ are both algebraic integers...

I've tried adding, subtracting and multiplying and fiddling around with the given alg. integers, and even got maple to do some of the work for me, but to no avail... Any ideas anyone?