Let $\alpha:=\sqrt[3]{17}$ and $K:=\mathbb{Q}(\alpha)$. We know that $$\mathcal{O}_K=\left\{\frac{a+b\alpha+c\alpha^2}{3}:a\equiv c\equiv -b\pmod{3}\right\}.$$ I have to show that $K$ has class number $1$, i.e. $\mathcal{O}_K$ is a PID. The Minkowski bound $\lambda <9$, so we should consider the primes $2, 3, 5, 7$. It's easy to show that
- $2\mathcal{O}_K=\mathfrak{p}_1\mathfrak{p}_2$, with $\mathfrak{p}_1=(2, \alpha+1)$ and $\mathfrak{p}_2=(2, \alpha^2+\alpha+1)$
- $3\mathcal{O}_K=\mathfrak{p}_3^2\mathfrak{p}_4$ (I can't compute these primes explicitly)
- $5\mathcal{O}_K=\mathfrak{p}_5\mathfrak{p}_6$, with $\mathfrak{p}_5=(5, \alpha+2)$ and $\mathfrak{p}_6=(5, \alpha^2+3\alpha-1)$
- $7\mathcal{O}_K=\mathfrak{p}_7$
Now, how can I show that, for example, $\mathfrak{p}_1$ and $\mathfrak{p}_5$ are principal ideals? (I can't find elements with norm $2$ or $5$). The situation for the prime $3$ is more complicated: the book suggests to find elements in $\mathcal{O}_K$ with norm $3$ that are coprime (this implies that $\mathfrak{p}_3$ and $\mathfrak{p}_4$ are principal), but I can't find these elements.
Note that $N_{K/\mathbb{Q}}(a+b\alpha+c\alpha^2)=a^3+17b^3+17^2c^3-3\cdot 17 abc$.
The first thing to do is get a clean basis without annoying congruence conditions : put $\beta=\frac{\alpha^2-\alpha+1}{3}$. Then $[1,\alpha,\beta]$ is a $\mathbb Z$-basis of $\mathcal{O}_K$.
We apply Franz Lemmermeyer's method and look for elements of the form $x+y\alpha+z\beta$ with interesting norms and $x,y,z$ small.
A little inspection shows that
$$ N(2+\alpha+\beta)=N(2-\beta)=2, N(1+\alpha+\beta)=N(1-\alpha+\beta)=3, N(3-\alpha)=2 \times 5 $$
A few additional checks and computations from here then reveals that
$$ \begin{array}{lclcl} \mathfrak{p}_1 &=& (2+\alpha+\beta) &=& \Bigg( \frac{\alpha^2+2\alpha+7}{3}\Bigg) \\ \mathfrak{p}_2 &=& (2-\beta) &=& \Bigg( \frac{-\alpha^2+\alpha+5}{3}\Bigg) \\ \mathfrak{p}_3 &=& (1+\alpha+\beta) &=& \Bigg( \frac{\alpha^2+2\alpha+4}{3}\Bigg) \\ \mathfrak{p}_4 &=& (1-\alpha+\beta) &=& \Bigg( \frac{\alpha^2-4\alpha+4}{3}\Bigg) \\ \mathfrak{p}_5 &=& (\frac{3-\alpha}{2+\alpha+\beta}) &=& \Bigg( \frac{-2\alpha^2-\alpha+16}{3}\Bigg) \\ \mathfrak{p}_6 &=& (\frac{5(2+\alpha+\beta)}{3-\alpha}) &=& \Bigg( \frac{11\alpha^2+28\alpha+74}{3}\Bigg) \\ \end{array} $$