A density theorem and its application

We compute the density of primes represented by a special quadratic form in a fixed square residue class. Using this result and a new method introduced by Thaine we prove the fact that for a prime p > 3 congruent to 3 modulo 4, the component e(p+1)/2 of the p-Sylow subgroup of the ideal class group of ℚ(ζp) is trivial.