Non-trivial facts about primes coming out of Algebraic Number Theory [closed]

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.

-

closed as not a real question by Dan Petersen, Marc Palm, Bruce Westbury, Chris Godsil, Henry CohnMar 21 '12 at 14:10

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Define "cool" :) – Qfwfq Mar 21 '12 at 10:39
How about: Which primes are a sum of two squares? – S. Carnahan Mar 21 '12 at 11:39
@Tiez "RE: For any prime p, p+1 is composite" is this true for p=2? – joro Mar 21 '12 at 12:21
@Tiez: please tell us your background in algebraic number theory already, so answers will have a better chance of being useful to you. That the first/only example you could come up with is that $p+1$ is (often) composite when $p$ is prime is unsettling because that's too trivial to make a good impression. It'd be like someone asking for applications of calculus and pointing out that he/she knows constant functions have derivative 0. – KConrad Mar 21 '12 at 13:19
@Tiez: Closing a question doesn't necessarily indicate that the question cannot be fixed, just that it needs some work and answers should be postponed until the question is clarified or improved. (And the "not a real question" is just because the software, which we can't change, forces people to choose from a limited set of reasons.) One option is just to try editing the question and see whether people vote to re-open it. Another is to start a discussion at tea.mathoverflow.net about what you are hoping to learn, whether it is appropriate for MO, and how to formulate the question. – Henry Cohn Mar 21 '12 at 15:53

Algebraic number theory solves the ancient/long-standing problem of providing a proof of quadratic reciprocity that those of us who are not Gauss can actually remember. Let p be an odd prime, and let K be the field obtained from Q by adjoining a primitive pth root of 1. Then K contains a unique quadratic extension of Q, which one sees easily is that obtained by adjoining a square root of p or -p according as p is congruent to 1 mod 4 or not. Now let q be a second odd prime. By computing the action of the Frobenius at q on the unique quadratic subfield of K in two different ways, one obtains the main statement of quadratic reciprocity.

-
@Anon, thank you(there are about 35 anon's on MO). I can follow your argument until the "two different ways". Can you please explain? – TZE Mar 21 '12 at 15:28
@Tiez: It guess it is this proof: en.wikipedia.org/wiki/… . – Emil Jeřábek Mar 21 '12 at 17:20
@Emil, thanks. I did not expect it will be that delicate. – TZE Mar 21 '12 at 21:06

In Cox, Primes of the form $x^2+ny^2$, you will find many examples.

Another example is the Chebotarev density theorem, of which Dirichlet's theorem on primes in arithmetic progressions is a special case.

-
@Timo Keller, thanks. – TZE Mar 21 '12 at 15:36