How to show that a number isn't an element of the ring of algebraic integers

Question

I have $\alpha = 2^{1/3}$ , $K = \mathbb{Q}(\alpha)$ a number field and $O_{K} = \mathbb{Z}(\alpha)$, the ring of algebraic integers in $K$. I want to be able to show that no number of the form $\frac{x + y \alpha + z \alpha^{2}}{2}$ can be in $O_K$, where $x , y , z \in \{0 , 1 \}$ are not all zero. I know that I might want to use the norm and/or the complex embeddings in some way but I'm really not sure how to go about it (even though I feel like the answer will be obvious).
I know there are a lot of ways to show that an element is in $O_{K}$, but I'm not sure how to show that it's not.

Answer 1

The following method works only for this special cute case:

Fact 1: $\frac{\alpha^2}{2}$ is not integral, by looking at its minimal polynomial.

Fact 2: If $\beta := \frac{x + y\alpha + z\alpha^2}{2}$ is integral, then $\alpha\beta = z + \frac{x\alpha + y\alpha^2}{2}$ is integral, and so is $\frac{x\alpha + y\alpha^2}{2}$, and so is $\frac{x\alpha^2}{2}$ by similar argument. Hence $x = 0$, and similar arguments show $y=z=0$.