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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.
11Y40 Algebraic number theory computations
11R29 Class numbers, class groups, discriminants
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[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 .
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