What is the norm of a number in a cubic integer ring? Let's say $K = \mathbb{Q}(\root 3 \of {10})$, and then $\mathcal{O}_K$ is a ring of algebraic integers. My calculations suggest numbers of the form $$\frac{a}{3} + \frac{a \root 3 \of {10}}{3} + \frac{a (\root 3 \of {10})^2}{3}$$ with $a \in \mathbb{Z}$ are algebraic integers. I imagine a few other forms are also algebraic integers, but I haven't been able to discover the pattern. Maybe congruence modulo $3$? Such as in $$\frac{5}{3} + \frac{11 \root 3 \of {10}}{3} + \frac{8 (\root 3 \of {10})^2}{3},$$ which has a minimal polynomial of $x^3 - 5x -285x + 1905$, unless I made a mistake somewhere along the way.
How do I compute the norms of numbers in $\mathcal{O}_K$? Is there a way to grab it off the minimal polynomial in a manner similar to quadratic integer rings?

If $\mathcal{O}_K$ with $K = \mathbb{Q}(\root 3 \of d)$ where $d > 1$ is a squarefree integer, then the norm of a number $a + b \root 3 \of d + c(\root 3 \of d)^2$ is $a^3 + b^3 d + c^3 d^2 - 3abcd$.
So then for example, the norm of $$\frac{5}{3} + \frac{11 \root 3 \of {10}}{3} + \frac{8 (\root 3 \of {10})^2}{3}$$ (which has a minimal polynomial of $x^3 - 5x^2 - 285x + 1905$, by the way) is $$\left(\frac{5}{3}\right)^3 + \left(\frac{11}{3}\right)^3 10 + \left(\frac{8}{3}\right)^3 100 - 3 \left(\frac{5}{3}\right)\left(\frac{11}{3}\right)\left(\frac{8}{3}\right)100$$ $$=\frac{21445}{9} - \left(\frac{440}{9}\right) 10 = 1905.$$
I found this answer in a question from 2013: Ring of integers of a cubic number field
