most recent 30 from http://mathoverflow.net 2013-05-22T23:16:13Z http://mathoverflow.net/feeds/question/91810 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91810/non-trivial-facts-about-primes-coming-out-of-algebraic-number-theory TZE 2012-03-21T10:15:43Z 2012-03-23T18:03:04Z

<p>What can be gleaned about primes from Algebraic Number Theory? I know this is too vague. What I mean is the following: </p> <p><strong>Are there several examples where Algebraic Number Theory helps to solve ancient/long-standing problems about primes?</strong></p> <p>Instances such as representibility of primes by quadratic forms <a href="http://en.wikipedia.org/wiki/Kaplanskys_theorem_on_quadratic_forms" rel="nofollow">1</a> and the quadratic reciprocity law <a href="http://en.wikipedia.org/wiki/Quadratic_reciprocity" rel="nofollow">2</a> have been suggested. <strong>What role does ANT play in the theory of prime numbers, specifically prime distribution, gaps and progressions?</strong> (Are there corresponding algebraic studies of these questions (in contract to the analytic point of view)? </p> <p>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.</p>

http://mathoverflow.net/questions/91810/non-trivial-facts-about-primes-coming-out-of-algebraic-number-theory/91821#91821 Timo Keller 2012-03-21T13:24:05Z 2012-03-21T14:04:38Z

<p>In Cox, Primes of the form $x^2+ny^2$, you will find many examples.</p> <p>Another example is the Chebotarev density theorem, of which Dirichlet's theorem on primes in arithmetic progressions is a special case.</p>

http://mathoverflow.net/questions/91810/non-trivial-facts-about-primes-coming-out-of-algebraic-number-theory/91822#91822 anon 2012-03-21T13:28:24Z 2012-03-21T13:28:24Z

<p>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.</p>