Representing the principal genus with prime ideals

Question

Let $K$ be a real quadratic number field. For an fractional ideal $\mathfrak a$ we denote with $[\mathfrak a]$ its narrow ideal class. Recall that $[\mathfrak a]=[\mathfrak b]$ holds if and only if there is a $\lambda \in K^\times$ with $\lambda \mathfrak a = \mathfrak b$ and $N(\lambda)>0$ (the norm condition makes the ideal class narrow).

Let $\mathfrak a$ be a fixed ideal belonging to the principal genus (i.e. $\mathfrak a= \lambda \mathfrak b^2$ for $\lambda \in K^\times$ with $N(\lambda)>0$ and a fractional ideal $\mathfrak b$). Are there infinitely many prime ideals $\mathfrak p \subset \mathcal O_K$ with $[\mathfrak a]=[\mathfrak p^2]$?

Answer 1

I say Chebotarev but all you need is that for a non-trivial finite order Hecke character the Hecke L-function $L(s,\psi)$ is analytic and non-zero at $s=1$, and for the trivial Hecke character we get the Dedekind zeta function which has a simple pole.

With $Cl^+(K)$ the narrow class group, each $\psi\in Hom(Cl^+(K),\Bbb{C}^\times)$ is a Hecke character, and $$\prod_{\psi\in Hom(Cl^+(K),\Bbb{C}^\times)} L(s,\psi)^{\psi(\mathfrak{b})^{-1}} = \exp(|Cl^+(K)| \sum_{\mathfrak{p}^k, [\mathfrak{p}^k] =[\mathfrak{b}]}\frac{N(\mathfrak{p}^k)^{-s}}{k})$$ has a pole at $s=1$ (due to the trivial character), which implies that there are infinitely many primes in the narrow class $[\mathfrak{b}]$, whose square will be in the narrow class $[\mathfrak{a}]=[\lambda \mathfrak{b}^2]$.