Constructing class fields over local fields

Pauli, Sebastian

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 627-652.

Soit $K$ un corps $𝔭$-adique. Nous donnons une caractérisation explicite des extensions abéliennes de $K$ de degré $p$ en reliant les coefficients des polynômes engendrant les extensions $L/K$ de degré $p$ aux exposants des générateurs du groupe des normes $N_{L/K}\left(L^{*}\right)$. Ceci est appliqué à un algorithme de construction des corps de classes de degré $p^m$, ce qui conduit à un algorithme de calcul des corps de classes en général.

Let $K$ be a $𝔭$-adic field. We give an explicit characterization of the abelian extensions of $K$ of degree $p$ by relating the coefficients of the generating polynomials of extensions $L/K$ of degree $p$ to the exponents of generators of the norm group $N_{L/K}\left(L^{*}\right)$. This is applied in an algorithm for the construction of class fields of degree $p^m$, which yields an algorithm for the computation of class fields in general.

[Ama71] S. Amano, Eisenstein equations of degree p in a $𝔭$-adic field. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 1–21. | MR | Zbl

[BC95] W. Bosma, J.J. Cannon, Handbook of Magma functions. School of Mathematics, University of Sydney, Sydney, 1995.

[Coh99] H. Cohen, Advanced topics in computational number theory. Springer Verlag, New York, 1999. | MR | Zbl

[Fie99] C. Fieker, Computing class fields via the Artin map. Math. Comp. 70 (2001), 1293–1303. | MR | Zbl

[FV93] I. B. Fesenko, S. V. Vostokov, Local fields and their extensions. Translations of Mathematical Monographs, vol. 121, American Mathematical Society, 1993. | MR | Zbl

[Has80] H. Hasse, Number Theory. Springer Verlag, Berlin, 1980. | MR | Zbl

[HPP03] F. Hess, S. Pauli, M. E. Pohst, Computing the multiplicative group of residue class rings. Math. Comp. 72 (2003), no. 243, 1531–1548. | MR | Zbl

[Iwa86] K. Iwasawa, Local class field theory. Oxford University Press, New York, 1986. | MR | Zbl

[Kra66] M. Krasner, Nombre des extensions d’un degré donné d’un corps $𝔭$-adique Les Tendances Géométriques en Algèbre et Théorie des Nombres, Paris, 1966, 143–169. | Zbl

[MW56] R. E. MacKenzie, G. Whaples, Artin-Schreier equations in characteristic zero. Amer. J. Math. 78 (1956), 473–485. MR 19,834c | MR | Zbl

[Pan95] P. Panayi, Computation of Leopoldt’s p-adic regulator. Dissertation, University of East Anglia, 1995.

[PR01] S. Pauli, X.-F. Roblot, On the computation of all extensions of a p-adic field of a given degree. Math. Comp. 70 (2001), 1641–1659. | MR | Zbl

[Ser63] J.-P. Serre, Corps locaux. Hermann, Paris, 1963. | MR | Zbl

[Yam58] K. Yamamoto, Isomorphism theorem in the local class field theory. Mem. Fac. Sci. Kyushu Ser. A 12 (1958), 67–103. | MR | Zbl