Riv. Mat. Univ. Parma, Vol. 6, No. 2, 2015

Laura Capuano[1] and Ilaria Del Corso[2]

A note on upper ramification jumps in Abelian extensions of exponent p

Pages: 317-329
Received: 30 July 2015   
Accepted in revised form: 16 October 2015
Mathematics Subject Classification (2010): 11S15, 11S20, 11S31.
Keywords:Elementary Abelian p-extensions, upper ramification jumps, normic groups, class field theory.
Author address:
[1] : Scuola Normale Superiore, Piazza dei Cavalieri 7, 56127 Pisa, Italy
[2] : Università di Pisa, Dipartimento di Matematica, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Abstract: In this paper we present a classification of the possible upper ramification jumps for an elementary Abelian \(p\)-extension of a \(p\)-adic field. The fundamental step for the proof of the main result is the computation of the ramification filtration for the maximal elementary Abelian \(p\)-extension of the base field \(K\). This result generalizes [3, Lemma 9, p.286], where the same result is proved under the assumption that \(K\) contains a primitive \(p\)-th root of unity. To deal with this general case we use class field theory and the explicit relations between the normic group of an extension and its ramification jumps, and we obtain necessary and sufficient conditions for the upper ramification jumps of an elementary Abelian \(p\)-extension of \(K\).

References

[1] N. P. Byott and G. G. Elder, Biquadratic extensions with one break, Canad. Math. Bull. 45 (2002), 168-179. MR1904081
[2] J. W. S. Cassels and A. Fröhlich, (eds.), Algebraic number theory, Academic Press, London and New York 1967. Zbl 0153.07403
[3] I. Del Corso and R. Dvornicich, The compositum of wild extensions of local fields of prime degree, Monatsh. Math. 150 (2007), 271-288. MR2309533
[4] G. G. Elder and J. J. Hooper, On wild ramification in quaternion extensions, J. Théor. Nombres Bordeaux 19 (2007), 101-124. MR2332056
[5] I. B. Fesenko and S. V. Vostokov, Local fields and their extensions, Transl. Math. Monogr., 121, 2nd ed., American Mathematical Society, Providence, RI 2002. MR1915966
[6] G. Gras, Class field theory. From theory to practice, Springer Monographs in Mathematics, Springer-Verlag, Berlin 2003. MR1941965
[7] E. Maus, Relationen in Verzweigungsgruppen (German), J. Reine Angew. Math. 258 (1973), 23-50. MR0345936
[8] H. Miki, On the ramification numbers of cyclic p-extensions over local fields, J. Reine Angew. Math. 328 (1981), 99-115. MR0636198
[9] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd ed., Springer-Verlag, Berlin; PWN-Polish Scientific Publishers, Warsaw, 1990. MR1055830
[10] J.-P. Serre, Local fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin 1979. MR0554237
[11] Y. Sueyoshi, On ramification of p-extensions of p-adic number fields, Mem. Fac. Sci. Kyushu Univ. Sez. A 32 (1978), 199-204. MR0509316


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