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Fast computation of class fields given their norm group

Loïc Grenié — 2008

Journal de Théorie des Nombres de Bordeaux

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

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