Mathematics syllabus for the PhD entrance examination Summary: Mathematics syllabus for the PhD entrance examination in the Scuola Normale Superiore Algebra Elementary properties of integers. Division algorithm. Euclidean algorithm. Primes. Existence and uniqueness of prime decomposition. Binomial coeffi- cients. Congruences and their properties. Congruence classes. Euler function. Euler's theorem. The field of complex numbers. Absolute value and argument of a complex number, and its trigonometric form. Absolute value and argument of a product. Roots of a complex number. Permutations. Decomposition into cycles. Sign of a permutation. Groups and their properties. Normal subgroups. Quotients. Lagrange's the- orem. The symmetric and alternating groups. Group actions, with applications to the structure of finite groups. Sylow's theorems. Rings and ideals. Domains and fields. Euclidean fields. Principal ideals domains. Unique factorization domains. Gauss's lemma. Polynomials with coefficients in a field, and their arithmetic. Roots. Irre- ducible polynomials. Real irreducible polynomials. Eisestein's criterion for the Collections: Mathematics