cours structure algébrique

cours structure algébrique

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Structure de groupe

Les trois structures algébriques que nous présentons maintenant (à savoir les structures de groupe, d'anneau et de corps) sont fondamentales. Commençons par la définition d'un groupe.

Structure d'anneau

Il est possible de munir un groupe (E, T) d'une seconde loi de composition interne. Si cette seconde loi possède de « bonnes » propriétés calculatoires, on obtient une structure algébrique appelée anneau. Notons * cette seconde loi de composition interne. Le triplet ( E, T, *) est encore appelé ensemble structuré.

Structure de corps

Dans un anneau, un élément ne possède pas nécessairement çle symétrique par rapport à la seconde loi (ou d'inverse puisque la seconde loi est souvent notée multiplicativement) . Il est à noter que le zéro d'un anneau est absorbant pour la seconde loi. Il ne peut donc pas être symétrisable pour la seconde loi.

Le corps des réels

Généralités On suppose connues les propriétés de l'ensemble N des entiers naturels et de l'ensemble Z des entiers relatifs. On désigne par + et x l'addition et la multiplication entre entiers.

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