I am trying to give an explicit expression of some ideal class groups of cubic number fields. I taking as a reference for the examples Number Fields by Daniel A. Marcus, in particular the cubic field on exercise 26, section 5, $\mathbb{Q}(\sqrt[3]{19})$. In this exercise, the author claims that
$R = \mathbb{A} \cap \mathbb{Q}(\sqrt[3]{19}) = \{\frac{a + b\alpha + c\alpha^{2}}{3} : a \equiv b \equiv c$ (mod 3)$\}$, with $\alpha = \sqrt[3]{19}$,
but I don't understand what leads him to this statement. I know that a basis of $R$ is $\{$ 1, $\alpha$, $\frac{\alpha^{2} \pm \alpha + 1}{3} \}$, but I don't know how to reach the result in question.
Also, in the previous exercise, ex.: 25, section 5, it says that for the cubic field $\mathbb{Q}(\sqrt[3]{17})$,
$R = \mathbb{A} \cap \mathbb{Q}(\sqrt[3]{17}) = \{\frac{a + b\alpha + c\alpha^{2}}{3} : a \equiv c \equiv -b$ (mod 3)$\}$, with $\alpha = \sqrt[3]{17}$,
and also I can't figure out why.