# Prime ideals and class group equations

Let $$K$$ be a degree $$n \geq 3$$ extension over $$\mathbb{Q}$$, and let $$\mathcal{O}_K$$ be its ring of integers. We say a rational prime $$p$$ splits completely in $$\mathcal{O}_K$$ if the principal ideal $$(p) = \mathfrak{p}_1 \cdot \cdots \cdot \mathfrak{p}_n$$ for pairwise distinct degree one prime ideals $$\mathfrak{p}_i$$, $$1 \leq i \leq n$$ (indeed, we do not want $$p$$ to ramify in $$\mathcal{O}_K$$). Let $$C_K$$ be the ideal class group of $$\mathcal{O}_K$$. For an ideal $$I \subset \mathcal{O}_K$$, let $$[I]$$ denote the corresponding ideal class in the class group.

Let $$g_1, \cdots, g_{n-1}$$ be arbitrary (not necessarily distinct) elements of $$C_K$$. Does there necessarily exist a rational prime $$p$$ which splits completely in $$\mathcal{O}_K$$ such that the ideal class equations $$g_i = [\mathfrak{p}_i]$$ holds for $$1 \leq i \leq n-1$$? Note that there are at most $$n-1$$ degrees of freedom since the product of the $$\mathfrak{p}_i$$'s is principal.

No. If $$K$$ is Galois, since the Galois group act transitively on the $$\mathfrak{p_i}$$, their class must be the same up to automorphism of the class group. Thus you can take as a counterexample any Galois extension $$K$$ of degree at least $$3$$ with nontrivial class group, $$g_1$$ the identity and $$g_2$$ any nonidentity element.