id: 05231679
dt: j
an: 05231679
au: Valibouze, Annick
ti: On a relation between the roots of a polynomial. (Sur les relations entre
    les racines d’un polynôme.)
so: Acta Arith. 131, No. 1, 1-27 (2008).
py: 2008
pu: Instytut Matematyczny PAN,Warszawa
la: FR
cc: 
ut: splitting field; Galois ideal; Galois group; maximal ideal
ci: Zbl 0943.12001
li: doi:10.4064/aa131-1-1
ab: Let $\overline{k}$ be an algebraic closure of a perfect field $k$ and let
    $\underlineα=(α_1,\ldots,α_n)\in \overline{k}^n$ be a tuple
    containing all the roots of a monic separable polynomial $f\in k[X]$ of
    degree $n$. In the polynomial ring $k[X_1,\ldots,X_n]$, one defines the
    ideal of $\underlineα$-relations as the set ${\germ M}$ of those
    $p$’s such that $p(\underlineα)=0$. It is a maximal ideal whose
    residue field $k[X_1,\ldots,X_n]/{\germ M}$ is $k$-isomorphic to the
    splitting field $k(\underlineα)$ of $f$. The decomposition group of
    ${\germ M}$, that is, the group of permutations of $\{X_1,\dots,X_n\}$
    which leave ${\germ M}$ invariant, corresponds to the Galois group $G$
    of $f$ as a group of permutations of $\{α_1,\ldots,α_n\}$.  This
    paper deals with the following problem: given $f$ as above, one wants
    to effectively compute $G$ and the reduced (lexicographic) Gröbner
    basis ${\germ I}$ of ${\germ M}$. The present paper is an extension of
    two unpublished works by the author, and a joint work with {\it S.
    Orange} and {\it G. Renault} (in revision in Exp. Math.). The
    author’s main goal is to compute ${\germ I}$ in a more efficient way
    than in previously known methods, including her own algorithm {\sl
    GaloisIdéal} [{\it A. Valibouze}, Bull. Belg. Math. Soc. Simon Stevin
    6, 507‒535 (1999; Zbl 0943.12001)]. She describes a new algorithm and
    gives some examples to illustrate the gains obtained with her method.
rv: Bernat Plans (Barcelona)