 
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
