Classic and extended Euclidean algorithm. Lehmer’s algorithm. Chinese theorem on remainders. Groups(Z/nZ) *. Legendre-Jacobi-Kronecker symbol. Square root modulo p. Solving polynomial equations modulo p. Algorithms for square matrices. Algorithms for general matrices. Z-modules. Hermite and Smith normal form. Grids and quadratic forms. Gram-Smith orthogonalization. Algorithms for reduction of grids. LLL algorithm. Algorithms over polynomials. Euclidean algorithm for polynomials. Factorization of polynomials modulo p. Factorization of polynomial sover Q or Z. Algebraic numbers and fields of digits. Track norm and characteristic polynomial. Discriminants and reduction of a polynomial. The algorithms of quadratic fields. Calculating Galois group. Elliptic curves. Factorization. Testing primes. Lehman’s, Pollard’s and Shanks’s method. Jacobi test. The method of elliptic curves. Software to support the theory of numbers.