TAILIEUCHUNG - Đề tài " The density of discriminants of quartic rings and fields "

Steiner symmetrization, one of the simplest and most powerful symmetrization processes ever introduced in analysis, is a classical and very well-known device, which has seen a number of remarkable applications to problems of geometric and functional nature. Its importance stems from the fact that, besides preserving Lebesgue measure, it acts monotonically on several geometric and analytic quantities associated with subsets of Rn. Among these, perimeter certainly holds a prominent position. Actually, the proof of the isoperimetric property of the ball was the original motivation for Steiner to introduce his symmetrization in. | Annals of Mathematics The density of discriminants of quartic rings and fields By Manjul Bhargava Annals of Mathematics 162 2005 1031 1063 The density of discriminants of quartic rings and fields By Manjul Bhargava 1. Introduction The primary purpose of this article is to prove the following theorem. Theorem 1. Let N i f ri denote the number of S4-quartic fields K having 4 2i real embeddings such that f Disc K n. Then a N4 x X _ ả n 1 p 2 p-3 p-4 X -tt X 48 1 _ p b Jim 4 X n 1 p-2 p-3 p-4 X -tt X 8 p c JimN4 0 X Ifi n i p-2 p-3 p-4 . X x X 16 p Several further results are obtained as by-products. First our methods enable us to count all orders in S4-quartic fields. Theorem 2. Let M i f n denote the number of quartic orders O contained in S4-quartic fields having 4 2i real embeddings such that f Disc O g. Then a lim X m40 0 X X _ z 2 2z 3 48 z 5 b lim m41 X 0 _ z 2 2z 3 X X 8 z 5 c lim m42 0 X _ z 2 2z 3 X X 16 z 5 Second the proof of Theorem 1 involves a determination of the densities of various splitting types of primes in S4-quartic fields. If K is an S4-quartic field unramified at a prime p and K24 denotes the Galois closure of K then the 1032 MANJUL BHARGAVA Artin symbol K24 p is defined as a conjugacy class in S4 its values being e 12 123 1234 or 12 34 where x denotes the conjugacy class of x in S4. It follows from the Chebotarev density theorem that for fixed K and varying p unramified in K the values e 12 123 1234 and 12 34 occur with relative frequency 1 6 8 6 3. We prove the following complement to Chebotarev density Theorem 3. Let p be a fixed prime and let K run through all S4-quartic fields in which p does not ramify the fields being ordered by the size of the discriminants. Then the Artin symbol K24 p takes the values e 12 123 1234 and 12 34 with relative frequency 1 6 8 6 3. Actually we do a little more we determine for each prime p the density of quartic fields K in which p has the various possible ramification types. For it. follows from .

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