What can be gleaned about primes from Algebraic Number Theory? I know this is too vague. What I mean is the following:
Are there several examples where Algebraic Number Theory helps to solve ancient/long-standing problems about primes?
Instances such as representibility of primes by quadratic forms 1 and the quadratic reciprocity law 2 have been suggested. What role does ANT play in the theory of prime numbers, specifically prime distribution, gaps and progressions? (Are there corresponding algebraic studies of these questions (in contract to the analytic point of view)?
I would be grateful if you point me to a survey on such topics. It doesn't hurt if the answer is No/None/Nothing, etc. Thanks.