NUMBER THEORY 1

Second cycle degree in MATHEMATICS

Language: English

Teaching period: First Semester

Lecturer: FRANCESCO BALDASSARRI

Number of ECTS credits allocated: 8

Syllabus
 Prerequisites: A standard Basic Algebra course; a short course in Galois Theory would be most useful; basic Linear Algebra; a basic course of Calculus; some familiarity with the theory of analytic functions of one complex variable would be useful. Examination methods: We will propose the preparation of 2 or 3 written reports during the course. These are supposed to check the step-by-step understanding of the topics presented and the interest of the students in the subject. A final all-inclusive exam will be proposed for those who have not presented satisfactory reports during the year as well as to those who are not satisfied with the mark obtained. Students will be offered to present one topic agreed with the teacher in a 45 minutes lecture during the course. A final oral examination is reserved for those who aim at top grades. Course unit contents: 1. Basic algebra of commutative groups and rings. 2. Factorization of elements and ideals 3. Dedekind domains 4. Algebraic number fields. Cyclotomic and quadratic fields. 5. Rings of integers. Factorization properties. 6. Finite extensions, decomposition, ramification. Hilbert decomposition theory. 7. Frobenius automorphism, Artin map; 8. Quadratic and cyclotomic fields. Quadratic reciprocity law. Gauss sums. 9. An introduction to Class Field Theory (from Kato-Kurokawa-Saito Vol. 2, Chap. 5) 10. Minkowski Theory (finiteness of class number and the unit theorem). 11. Dirichlet series, zeta function, special values and class number formula. The whole material is to be found in the single textbook: Daniel A. Marcus "Number Theory", Springer-Verlag. The essential part of the program consists of Chapters 1 to 5, with those exercises which are used in the body of the textbook. Chapters 6 and 7 are required to get a higher grade. The lengthy real-analytic proofs in Chapters 5/6/7 are not essential. A good understanding of the complex-analytic strategy is necessary. We recommend, for cultural reasons, reading through the two volumes of Kato-Kurokawa-Saito, possibly without studying proofs.