I am working on a senior thesis, and my advisor told me to look into the theory that prime ideals in a number field of norm less than $N$ are evenly distributed across ideal classes.
I've looked at the Stevenhagen article here , and it says that the desired result is a consequence of applying Chebotarev's Density Theorem to a well-chosen polynomial, and requires applying the Hilbert class field. However, I have struggled finding resources that explain this, and, although I have studied algebraic number theory (I took a class for which the textbook was Samuel's Algebraic Theory of Numbers), I don't have a strong background in class field theory. I don't even understand how Stevenhagen got that you could get the degrees of the irreducible factors of $x^{10}-1 (\text{mod }p)$ based on the residue of $p$ mod 10, apart from being able to get the number of monic factors from the number of elements of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})$ that satisfy $u^{10}\cong 1 (\text{mod }p)$.
Any resources that you suggest I look at? Thank you so much!
You don't need class field theory to prove that Hecke L-functions $L(s,\phi)$ where $\phi$ is a character of the ideal class group have an analytic continuation and so on, making them similar to Dirichlet L-functions, so that the same proof as the PNT in arithmetic progressions works, giving that prime ideals are equidistributed in ideal classes.
This PNT for Hecke L-functions is part of the proof of Chebotarev's theorem.