Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-29T13:41:21.864Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit local reciprocity for tame extensions

Published online by Cambridge University Press:  21 December 2011

RACHEL NEWTON
RACHEL NEWTON*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA. e-mail: rdn21@cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a tamely ramified abelian extension of local fields of degree n, without assuming the presence of the nth roots of unity in the base field. We give an explicit formula which computes the local reciprocity map in this situation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 3 , May 2012 , pp. 425 - 454
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Artin, A. and Hasse, H.Die beiden Ergänzungsätze zum Reziprozitätsgesetz der l n-ten Potenzreste in Körper der l n-ten Einheitswurzeln. Abh. Math. Sem. Univ. Hamburg 6 (1928), 146–162.CrossRefGoogle Scholar
[2]Cassels, J. W. S. and Fröhlich, A. (eds.). Algebraic Number Theory (Academic Press, 1967).Google Scholar
[3]Coates, J. and Wiles, A.Explicit reciprocity laws. Soc. Math. France, Astérisque 41–42 (1977), 7–17.Google Scholar
[4]Gille, P. and Szamuely, T.Central Simple Algebras and Galois Cohomology (Cambridge University Press, 2006).CrossRefGoogle Scholar
[5]De Shalit, E.The explicit reciprocity law in local class field theory. Duke Math. J. 53, Number 1 (1986), 163–176.CrossRefGoogle Scholar
[6]Fesenko, I. B. and Vostokov, S. V.Local Fields and Their Extensions: Second Edition (American Mathematical Society, 2002).Google Scholar
[7]Greenberg, M. J.An elementary proof of the Kronecker-Weber Theorem. Amer. Math. Monthly 81 (1974), 601–607; corr.: ibid. 82 (1975), 803.CrossRefGoogle Scholar
[8]Hasse, H.Die Normenresttheorie relativ-Abelscher Zahlkörper als Klassenkörpertheorie im Kleinen. J. Reine Angew. Math. (Crelle) 162 (1930), 145–154.CrossRefGoogle Scholar
[9]Hasse, H.Über -adische Schiefkörper und ihre Bedeutung für die Arithmetik hyperkomplexer Zahlensysteme. Math. Ann. 104 (1931), 495–534.CrossRefGoogle Scholar
[10]Iwasawa, K.On explicit formulas for the norm residue symbol. Math. Soc. Japan 20 (1968), 151–164.CrossRefGoogle Scholar
[11]Milne, J. S.Class Field Theory. online notes (v4.00), www.jmilne.org/math/, 2008.Google Scholar
[12]Roquette, P.The Brauer–Hasse–Noether theorem in historical perspective. Schriften der Mathematisch-naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften 15 (Springer, 2005).Google Scholar
[13]Schmidt, F. K.Zur Klassenkörpertheorie im Kleinen. J. Reine Angew. Math. (Crelle) 162 (1930), 155–168.CrossRefGoogle Scholar
[14]Serre, J.-P.Local Fields. Graduate Texts in Mathematics 67 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[15]Wiles, A.Higher explicit reciprocity laws. Ann. of Math. 107 (1978), 235–254.CrossRefGoogle Scholar