Number fields. Number rings. Discriminant for a number field. Fractional ideals. Unique factorization for fractional ideals in a number ring. Class group. Minkowski's theorem. Finiteness of the class group. Dirichlet's units theorem. First Case of Fermat's Last Theorem for regular primes. Dirichlet's L-functions. Riemann's zeta function. Dirichlet's theorem. p-adic numbers. Quadratic forms over the p-adics. Integer quadratic forms.
Stewart-Tall: Algebraic number theory. Chapman and Hall Mathematics Series Chapman & Hall, London, 1987.
Marcus: Number Fields Universitext. Springer-Verlag, New York-Heidelberg, 1977
Serre : A course in arithmetic. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973
Learning Objectives
1 Knowledge acquired:
The aim of the course is to give the fundamental knowledge of algebraic number theory . In a second stage the tools developed are used to prove some important results, e.g. Dirichlet's units theorem, the finiteness of class groups etc…
Competence acquired:
The student who has successfully passed the exam , should have the basic knowledge to understand more advanced topics in number theory.
Skills acquired (at the end of the course):
The student should be able to perform calculations in number fields, manipulating elements as well as ideals.
Prerequisites
Courses required: Algebra I,II
Teaching Methods
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 150
Hours reserved to private study and other indivual formative activities: 102
Hours for lectures: 48
Students are free to go in the professor's study when they need clarification or explanation, otherwise they can ask for an appointment.
Type of Assessment
Oral exam
Course program
Downloadable from http://web.math.unifi.it/users/puglisi/didattica.html
Number fields. Number rings. Discriminant for a number field. Fractional ideals. Unique factorization for fractional ideals in a number ring. Class group. Minkowski's theorem. Finiteness of the class group. Dirichlet's units theorem. First Case of Fermat's Last Theorem for regular primes. Dirichlet's L-functions. Riemann's zeta function. Dirichlet's theorem. p-adic numbers. Quadratic forms over the p-adics. Integer quadratic forms.