By Dumas J.-G.

Summary: In each fi eld of scientifi с and commercial study, the extension of using desktop technological know-how has led to an expanding desire for computing energy. it really is therefore important to take advantage of those computing assets in parallel. during this thesis we search to compute the canonical kind of very huge sparse matrices with integer coeffi cients, particularly the integer Smith common shape. through 'Very large'', we suggest 1000000 indeterminates and one million equations, i.e. thousand billion of coeffi cients. these days, such platforms usually are not even storable. even though, we're attracted to platforms for which a lot of those coeffi cients are exact; for that reason we discuss sparse structures. we wish to clear up those platforms in a precise manner, i.e. we paintings with integers or in smaller algebraic constructions the place all of the easy mathematics operations are nonetheless legitimate, particularly fi nitefi elds. The rebuilding of the complete answer from the smaller suggestions is then quite effortless.

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Sorties : – Un polynôme irréductible de ✁✡✂ ☎✞✝ ✟ ✕ ☞ ✗ , de degré ✠ . Pour tout ✆ ✁ ✁✡✂☛☎✞✝ ✟ ✆ ✝ ✄ Faire ✂ ✠ Si Test-Irréductibilité ☎ ☞ ✟ ✠ ✆ ✟ == “Oui” Alors Renvoyer ☞ ✟ ✠ ✆ ✠☛✞ ✓ Faire Pour ✠ ✠Pour☞☛ toutjusqu’à ✆ ✁ ✁✡✂ ☎✞✝ ✟ ✂ ✆ ✠ ✝ ✄ ✂ ✁✞✁ ✁✄✂☛☎✞✝ ✟ ✂ ✁ ✠ ✝ ✄ Faire ✠ ✆ ✟ == “Oui” Alors Renvoyer ☞ ✟ ✠ Si Test-Irréductibilité ☎ ☞ ✟ ✠ ✁ ☞ ✁☞ ✠ ✆ ✓ ☞ ✌✏✎ ✠ ☎✑✡ ✂ ☞ ✟ ✠ ✓ et ✟ ☎✑✡ ✟ minimal tel que l’ordre de ☞ dans ✓✒ Soit ✡☞☛ ☞ ✟ ✞ , avec ✍ soit ✠ . Si ✟ ☎✑✡ ✟ est petit Alors Entrées : – – Un entier ✠ 1 : 2 : 3 : 4 : ☞ 5 : ☞ 6 : n 7 : 8 : Renvoyer un facteur de Algorithme E] } ✔ ✁ ☎☞ ✟ {Par [92 - Kaltofen et Shoup (1997), 9 : Sinon 10 : 11 : Répéter Sélectionner Jusqu’à ce que Test-Irréductibilité ☎ 13 : Renvoyer ✄ 12 : .

4 Construction des corps de Galois . . . . Racines primitives dans les sous-groupes de Générateurs des extensions . . . . . 1 Polynômes cyclotomiques . . . . 3 Racines primitives creuses de ✁✡✂☛☎✞✝ ✟✠✟ . Efficacité de l’☞ -Irréductibilité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 47 50 51 52 54 56 42 Construction des corps finis Rappelons tout d’abord quelques définitions et propriétés des corps finis que nous utiliserons par la suite.

43 47 50 51 52 54 56 42 Construction des corps finis Rappelons tout d’abord quelques définitions et propriétés des corps finis que nous utiliserons par la suite. 1 – Un corps est un anneau non vide dans lequel tout élément non nul possède un inverse. L’ensemble des inversibles d’un ensemble est noté . ✄✂ – Le corps des nombres rationnels ✁ et les corps p , pour ✏ premier, sont appelés corps premiers. – Les corps finis sont appelés corps de Galois. Ils sont notés ✁✄✂✆☎✞✝✠✡ , où ✝ est le cardinal du corps.

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