An efficient algorithm for the computation of Galois automorphisms

Abstract

Abstract. We describe an algorithm for computing the Galois automorphisms of a Galois extension which generalizes the algorithm of Acciaro and Klüners to the non-Abelian case. This is much faster in practice than algorithms based on LLL or factorization.

Explicit determination of generalized symmetric and alternating Galois groups

“…(1), the polynomial F τ is computed. The procedure is similar to that given by Allombert [2] to determine diagonal automorphisms.…”

Topics in computational algebraic number theory

“…In many applications, it is advantageous to use non-Archimedean embeddings K → K ⊗ Q Q p = ⊕ p|p K p which is isomorphic to Q n p as a Q p -vector space. This cancels rounding errors, as well as stability problems in the absence of divisions by p. In some applications (e.g., automorphisms [1], factorization of polynomials [3,18]), a single embedding K → K p is enough, provided an upper bound for α is available.…”

“…It is enough to prove that w 1 is never swapped with its size-reduced successor, say s. Let w * 1 = w 1 and s * be the corresponding orthogonalized vectors. A swap occurs if s * < w 1 c − µ 2 , where the Gram-Schmidt coefficient µ = µ 2,1 satisfies |µ| 1/2 (by definition of size-reduction) and s * = s − µw 1 . From the latter, we obtain…”

Une Famille Infinie d'Extensions Faiblement Ramifiées

“…Le deuxième algorithme utilisé, dûà Allombert [1] et implémenté dans PARI (fonction galoisinit), calcule explicitement les automorphismes d'une extension galoisienne de Q. En particulier, on peut s'assurer que les valeurs du paramètre considérées sont de bonnes valeurs.…”