Dirichlet’s theorem states that for each integer \(m > 1\) and each integer \(a\) coprime to \(m\), there are infinitely many primes \(p \equiv a \operatorname{mod}m\)
More is true: Chebotarev density 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 in \(L\), then in fact \(L = K\).