Summary

Résumé Structure Algébrique 1

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Course
Structure Algébrique

Institution
EPITA

Cette fiche résume le concept fascinant de la structure algébrique. En parcourant ses pages, vous découvrirez les fondements et les principes clés de l'algèbre, ainsi que les différentes structures qui en découlent. De la théorie des groupes à celle des anneaux et des corps, vous explorere...

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Uploaded on June 14, 2023
Number of pages 9
Written in 2021/2022
Type Summary

structure algébrique
algèbre
mathématiques
groupes
anneaux
corps
opérationsalgébriques
théoriedesgroupes
théoriedesanneaux
théoriedescorps
algèbreabstraite
loisalgébriques
algèbrelinéaire
algèbrecommu

Institution
EPITA
Education
Informatique
Course
Structure Algébrique

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Structure Algébrique

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1. Summary - Structure algébrique 1
2. Summary - Structure algébrique 2
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, Définition On toute abstraction
entend structure
algébrique
• :

par
des structures de calcul standart que vous avez pu rencontrer .

On va

I.
aborder

Anneaux
3 chapitres
factoriels
:

euclidiens et corps
}
, .

les
2.
Algèbre linéaire sur esp .
rect .
Sur un corps .

3. Groupes et
symétrie .

C'est étudier les relations de divisibilité entre entiers

Définition Divisibilité

}
sur Z
:

*
l
Un élément K E Z divise un entier m E Z S' il existe un entier C- Z
tq
m
Pour parler de division
• :

*
produit
→
addition ( + et * )

* entier se divise lui -

même ( 1 ✗ a
=
a x n )
*
que le diviseur de deux nomb diviseurs de la somme ( acb + c) =
ab tac )

1. La notion d' anneau

Définition

Un anneau est un triplet CA , +
,
_ ) composé
•
d' un ensemble A contenant deux éléments spéciaux 0 et 1
,

qu' on appelle resp .
élément neutre pour l' addition et éteint

neutre pour la multiplication .

•
et de 2 Opérations binaires A ✗ A → A + et ,
resp .
appelé addition
et multiplication .

qui satisfont pour l' addition .

{
( assiociativité) fa ,
b, C E A a + Cbt c) =
( a + b) + c
,

Groupe ( 7 neutre ) ta c- A
,
at o
=
☐ + a =
a

( inversibilité ) ta c- A ,
J C- a) C- A tq a + C- a) =
C- a) + a
=
0

Abélien →
( commutativité ) fa ,
b C- A ,
a + b =
b. + a

la multiplication
pour
( associativité ) fa , b, C C- A
,
a. ( b. c) =
(a .

b) •
C

( J neutre ) ta c- A a. 1
=
1
.

a
=
a ( 0A # 1A )
,

1.

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