Dirichlet’s theorem states that for each integer $m > 1$ and each integer $a$ coprime to $m$, there are infinitely many primes $p \equiv a \mod m$

More is true: [Chebotarev density](Chebotarev%20density.md) tells us that for each modulus $m$ the primes are equidistributed among the residue classes of the integers $a$ coprime to $m$.

Dirichlet's theorem states that if $N\geq 2$ is an integer and a is coprime to N, then the proportion of the primes p congruent to a mod N is asymptotic to 1/n, where n=φ(N) is the Euler totient function. This is a special case of the Chebotarev density theorem for the Nth cyclotomic field K

t implies that as a Galois extension of $K$, $L$ is uniquely determined by the set of primes of K that split completely in it. 
A related corollary is that if almost all prime ideals of $K$ [split completely](split%20completely) in $L$, then in fact $L = K$.