Explicit bounds for primes in residue classes

Let $E/K$ be an abelian extension of number fields, with $E \ne \Bbb Q$. Let $\Delta$ and $n$ denote the absolute discriminant and degree of $E$. Let $\sigma$ denote an element of the Galois group of $E/K$. We prove the following theorems, assuming the Extended Riemann Hypothesis:

(1)

There is a degree-$1$ prime ${\frak p}$ of $K$ such that $\left (\frac {\displaystyle \frak p}{E/K}\right ) =\sigma$, satisfying $N{\frak p}\le (1+o(1))(\log \Delta+2n)^2$.

(2)

There is a degree-$1$ prime $\frak p$ of $K$ such that $\left (\frac {\displaystyle \frak p}{E/K}\right )$ generates
the same group as $\sigma$, satisfying $N{\frak p}\le (1+o(1))(\log \Delta)^2$.

(3)

For $K=\Bbb Q$, there is a prime $p$ such that $\left (\frac {\displaystyle \frak p}{E/\Bbb Q }\right )=\sigma$, satisfying
$p\le (1+o(1))(\log \Delta)^2$.

In (1) and (2) we can in fact take $\frak p$ to be unramified in $K/\Bbb Q$. A special case of this result is the following.

(4)

If $\gcd (m,q)=1$, the least prime $p\equiv m\pmod q$ satisfies
$p\le (1+o(1))(\varphi (q)\log q)^2$.

It follows from our proof that (1)--(3) also hold for arbitrary Galois extensions, provided we replace $\sigma$ by its conjugacy class $\langle \sigma\rangle$. Our theorems lead to explicit versions of (1)--(4), including the following: the least prime $p \equiv m \pmod q$ is less than $2( q \log q )^2$.