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Extra resources for Congruences, applications

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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.

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Congruences, applications by Pascal Boyer


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