×

zbMATH — the first resource for mathematics

Fast computation of class fields given their norm group. (English) Zbl 1193.11119
Summary: 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 \(\mathfrak m\). Using the vector-space structure of \(\text{Gal}(K_{\mathfrak m}/K)\), 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.
MSC:
11Y40 Algebraic number theory computations
11R29 Class numbers, class groups, discriminants
Software:
PARI/GP
PDF BibTeX XML Cite
Full Text: DOI Numdam Numdam EuDML
References:
[1] Henri Cohen, Advanced Topics in Computational Number Theory, volume 193 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. · Zbl 0977.11056
[2] Loïc Grenié, Comparison of semi-simplifications of Galois representations. J. Algebra 316 (2) (2007), 608-618. · Zbl 1193.11052
[3] The PARI Group, Bordeaux. PARI/GP, version 2.4.1, 2006. Available from .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.