#### Objective

#### Content

- Divisibility in the integral rings, factorial rings, principal ideal rings, arithmetic in Z and in K[X]: Congruence. Operations on the ideals in a commutative ring, Chinese remainder theorem. Arithmetic functions. Euler’s function. Structures of the group of invertible elements modulo n. Finite fields: Chevalley’s theorem, quadratic reciprocity law.
- Integral elements over a ring, integrally closed rings, Dedekind rings, field of numbers, ring of integers, ideals, norm of an ideal, ideal classes, examples of a quadratic fields and cyclotomic fields. Discrete subgroups of . Finiteness of the ideal class group of an algebraic number field, the unit theorem, units in a quadratic field. The splitting of a prime number in a number field, the splitting of a prime ideal in a field of numbers, discriminant and ramification. Kummer’s theorem. Galois extension of a field of numbers. Application on cyclotomic fields.
Continuous fractions, Quadratic rational forms, Diophantine equations: (Pell, Fermat, Mordel…). Applications. The two-squares theorem. The four-squares theorem.
[ Some details could be changed every year according to the research needs ].