Serre’s formule de masse in prime degree

详细信息    查看全文

• 作者：Chandan Singh Dalawat (1)
  
• 关键词：Formule de masse de Serre ; Serre’schen Ma?formel ; Serre’s mass formula ; 11S15 ; 11S20 ; 12F10
• 刊名：Monatshefte f眉r Mathematik
• 出版年：2012
• 出版时间：April 2012
• 年：2012
• 卷：166
• 期：1
• 页码：73-92
• 全文大小：324KB
• 参考文献：1. Artin E.: Galois Theory, pp. i+70. University of Notre Dame, Notre Dame (1942)
  2. Brown K.: Cohomology of Groups, pp. x+306. Springer, New York (1994)
  3. Cassels J.: Local Fields, pp. xiv+360. Cambridge University Press, Cambridge (1986)
  4. Dalawat C.: Local discriminants, kummerian extensions, and elliptic curves. J. Ramanujan Math. Soc. <strong class="a-plus-plus">25strong>(1), 25-0 (2010) (Cf. arXiv:0711.3878v2)
  5. Dalawat, C.: A first course in Local arithmetic (arXiv:0903.2615v1)
  6. Dalawat C.: Further remarks on local discriminants. J. Ramanujan Math. Soc. <strong class="a-plus-plus">25strong>(4), 391-17 (2010) (Cf. arXiv:0909.2541v1)
  7. Dalawat C.: Final remarks on local discriminants. J. Ramanujan Math. Soc. <strong class="a-plus-plus">25strong>(4), 419-32 (2010) (Cf. arXiv:0912.2829v2)
  8. Del Corso I., Dvornicich R.: The compositum of wild extensions of local fields of prime degree. Monatsh. Math. <strong class="a-plus-plus">150strong>(4), 271-88 (2007) ss="external" href="http://dx.doi.org/10.1007/s00605-006-0417-5">CrossRef
  9. Doud D.: Wild ramification in number field extensions of prime degree. Arch. Math. (Basel) <strong class="a-plus-plus">81strong>(6), 646-49 (2003) ss="external" href="http://dx.doi.org/10.1007/s00013-003-4722-0">CrossRef
  10. Hasse H.: Zahlentheorie, pp. 611. Akademie-Verlag, Berlin (1969)
  11. Neumann O.: Two proofs of the Kronecker–Weber theorem “according to Kronecker, and Weber- J. Reine Angew. Math. <strong class="a-plus-plus">323strong>, 105-26 (1981)
  12. Serre J.-P.: Corps locaux, Publications de l’Université de Nancago VIII, pp. 245. Hermann, Paris (1968)
  13. Serre J.-P.: Une “formule de masse-pour les extensions totalement ramifiées de degré donné d’un corps local. Comptes Rendus <strong class="a-plus-plus">286strong>, 1031-036 (1978)
  
• 作者单位：Chandan Singh Dalawat (1)
  
  1. Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad, 211019, India
  
• ISSN：1436-5081

文摘

For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered F p[G]-module K ?×?/K ?× p in characteristic 0 and ${K/\wp(K)}$ in characteristic p, where ${K=F(\root{p-1}\of{F^\times})}$ and G?= Gal(K|F). As an application, we give an elementary proof of Serre’s mass formula in degree p. We also determine the compositum C of all degree-p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is ${K(\root p\of{K^\times})}$ or ${K(\wp^{-1}(K))}$ , respectively, in the case of the local field F.