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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 $𝔪$and its norm group. Let $K_{𝔪}$ be the compositum of degree $\ell$ 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.