= Solution
The <Kummer-Dedekind theorem> states the following. If $K=\mathbb Q(\theta)$ with $\theta$ integral and $p\nmid[\mathcal O_K:\mathbb Z[\theta]]$, factor the <minimal polynomial> modulo $p$ as $\bar f=\prod_i\bar g_i^{e_i}$ into distinct monic irreducible factors. Then
$$
p\mathcal O_K=\prod_i\mathfrak p_i^{e_i},\qquad
\mathfrak p_i=(p,g_i(\theta)),\qquad
N(\mathfrak p_i)=p^{\deg g_i}.
$$
The index restriction is essential.
For the proposed <ideal> relation, reflexivity uses $\alpha=\beta=1$ and symmetry interchanges $\alpha,\beta$. If $(\alpha)I_1=(\beta)I_2$ and $(\gamma)I_2=(\delta)I_3$, multiplication gives $(\alpha\gamma)I_1=(\beta\delta)I_3$. All multipliers remain nonzero. Thus it is an <equivalence relation>, namely equality of <ideal classes>.
Put $s=\sqrt{-2}$. The <minimal polynomial> is $x^2+2$ and the given full power <basis> makes every prime eligible for the theorem. Modulo $3$ its roots are $\pm1$; modulo $5$ and $7$ it is irreducible; modulo $17$ its roots are $\pm7$. Consequently
$$
\boxed{\begin{aligned}
(3)&=(3,s-1)(3,s+1),\\
(5)&\text{ is prime},\\
(7)&\text{ is prime},\\
(17)&=(17,s-7)(17,s+7).
\end{aligned}}
$$
The two inert ideals have norms $25$ and $49$; each split factor has <norm> equal to its rational prime.
For $p_1\equiv1\pmod8$, <quadratic reciprocity> and its supplementary laws give $(-2/p_1)=1$. Thus $x^2+2$ has two distinct roots modulo $p_1$, giving distinct prime ideals $P,P'$ with
$$
(p_1)=PP',\qquad N(P)=N(P')=p_1.
$$
The given <principal ideal domain> property makes $P=(x+ys)$ for some integers $x,y$. Taking norms, which are positive here, gives $p_1=N(P)=x^2+2y^2$.
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