Hi there,


Here is a part of the book of Murty "an introduction to Sieve methods and applications"
page 36


"At this point we invoke some algebraic number theory let $K=Q(\theta)$ where $\theta$ is the solution of the polynomial $f(x)$. The ring of integers $O\_{k}$ of K is a Dedekind Domain. it's a classical theorem of Dedekind that for all but finitely many primes $\delta\_{f}(p)$ is the number of prime ideals $p$ of $O\_{k}$ such that the norm $N\_{K/Q}(p)=p$"


Note: $\delta\_{f}(p)$:= number of solutions of $f(x)$ modulo $p$


Question: What can be said about the size of the set of primes breaking this rule?
In my opinion, might they be those dividing $disc(f)$?


Yildo



PNT for number fields.
This statement is true for all $p$ not dividing $disc(f)$, as you say. Moreover you can write down exactly what the primes dividing $p$ are: they're the ideals $(p, g(\theta))$ for each irreducible factor $g$ of $f$. The norm of such a prime is the degree of $g$, so the number of degree 1 primes is the number of roots of $f$ mod $p$.


