By Pascal Boyer
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Extra resources for Congruences, applications
Example text
Soit maintenant a, b tels que b − a est divisible par le pgcd (n, m). On ´ecrit une relation de Bezout (b−a) n m un + vm = (n, m) et on pose k = u (n,m) b + v (n,m) b. On a alors k = un (n,m) + a ≡ a mod n ; (a−b) de mˆeme on a k = vm (n,m) + b ≡ b mod m, de sorte que l’ensemble donn´e est inclus dans l’image de π. La r´eciproque est ´evidente car k = a + λn = b + µm soit (b − a) = λn − µm qui est donc divisible par (n, m). En particulier lorsque n et m sont premiers entre eux, π induit un isomorphisme Z/nmZ Z/nZ × Z/mZ.
Cours d’alg`ebre. Ellipses, 1998. [6] P. Samuel. Th´eorie alg´ebrique des nombres. Hermann, 1967. -P. Serre. Cours d’arithm´etique. Puf, 1970. P. Smart. The algorithmic resolution of diophantine equations, volume 41. London Mathematical Society Student Texts, 1998.
P. Smart. The algorithmic resolution of diophantine equations, volume 41. London Mathematical Society Student Texts, 1998.
Congruences, applications by Pascal Boyer
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