Le théorème fondamental de l'arithmétique permet d'affirmer que tout entier supérieur
ou égal à 2 possède une décomposition en facteurs premiers. C'est-à-dire qu'il peut
s'écrire de manière unique comme le produit fini de nombres premiers à une puissance
adéquate.
Traduction
un idéal premier d'un anneau commutatif unitaire est un idéal tel que le quotient de
l'anneau par cet idéal est un anneau intègre.
les Anneaux noetheriens
Soit A un anneau commutatif. On dit qu’il est principal s’il est 1/ intègre
et 2/ si tout idéal de A est engendré par un élément.
Les Anneaux Factoriels.
l’elt irr ́eductible 3 n’est pas premier.
Anneax euclidiens
eisenstein
If a Euclidean domain is not a field then it has an element a with the following property:
any element x not divisible by a can be written as x=ay+u for some unit u and some element y.
This follows by taking a to be a non-unit with f(a) as small as possible.
This strange property can be used to show that some principal ideal domains are not Euclidean domains, as not all PIDs have this property.
For example, for d = −19, −43, −67, −163, the ring of integers of Q d
{\mathbb Q}({\sqrt {d}}) is a PID which is not Euclidean, but the cases d =
−1, −2, −3, −7, −11 are Euclidean.[7]
Irréductibles dans Z[i]
Irréductibles dans Z[i]
PGCD et PPCM
Existence de PGCD
Calculs de PGCD
Theoreme de Gauss
Corps de fractions définition, proprietes
Irréductibles dans A[X]
Contenu : existence, proprietes
Critères d’irréductibilité dans A[X]
Eisenstein's criterion to apply for a prime number
p it must divide both non-leading coefficients 15 and 10, which means only p = 5 could work, and indeed it does since 5 does not divide the leading coefficient 3,
and its square 25 does not divide the constant coefficient 10.
One concludes that Q is irreducible over the rationals and,since it is primitive, over Z as well
Note that since Q is of degree 4, this conclusion could not have been established by only checking that Q has no rational roots (which eliminates possible factors of degree 1), since a decomposition into two quadratic factors could also be possible.
