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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.
##### MSC:
 11Y40 Algebraic number theory computations 11R29 Class numbers, class groups, discriminants
PARI/GP
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##### 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 .
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