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Loïc Grenié
Fast computation of class fields given their norm group
Journal de théorie des nombres de Bordeaux, 20 no. 3 (2008), p. 707-714, doi: 10.5802/jtnb.646
Article PDF | Reviews MR 2523313 | Zbl pre05572697

Résumé - Abstract

Let $K$ be a number field containing, for some prime $\ell $, the $\ell $-th roots of unity. Let $L$ be a Kummer extension of degree $\ell $ of $K$ characterized by its modulus $\mathfrak{m}$and its norm group. Let $K_\mathfrak{m}$ be the compositum of degree $\ell $ extensions of $K$ of conductor dividing $\mathfrak{m}$. Using the vector-space structure of $\operatorname{Gal}(K_\mathfrak{m} / K)$, we suggest a modification of the rnfkummer function of PARI/GP which brings the complexity of the computation of an equation of $L$ over $K$ from exponential to linear.

Bibliography

[Coh] Henri Cohen, Advanced Topics in Computational Number Theory, volume 193 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000.  MR 1728313 |  Zbl 0977.11056
[Gre] Loïc Grenié, Comparison of semi-simplifications of Galois representations. J. Algebra 316 (2) (2007), 608–618.  MR 2356847 |  Zbl pre05222082
[PAR] The PARI Group, Bordeaux. PARI/GP, version 2.4.1, 2006. Available from http://pari.math.u-bordeaux.fr/.