Have awareness of the content and meaning of basic theoretical results relating to analytical and algebraic aspects of number theory, with particular regard to the distribution of prime numbers and the classification of number fields. Know how to consciously reproduce the main demonstrative phases of theory construction. Know how to frame and solve some exercises and problems on the main topics of the course, such as linear systems and quadratic equations of congruences, the classification of number fields.
Course Prerequisites
Courses of Linear Algebra, Algebra 1, Algebra 2 and Algebra 3
Teaching Methods
Lectures and excercise lessons
Assessment Methods
The exam consists of a written test, during which the student will solve some exercises, and an oral test, during which the student will answer some mainly theoretical questions. To access the oral exam, the student is required to achieve a minimum score of 15/30 in the written part. In the oral part we will focus above all on verifying the level of knowledge of the notions presented during the course, the clarity with which they are presented and the student's ability to apply them. The formulation of the grade will be obtained by considering the overall breadth and depth of learning, as well as the clarity of the presentation and the skills demonstrated in problem solving. The grade will be obtained from the comparison, not necessarily reduced to an arithmetic mean, of the evaluation of the written part and the oral part.
Texts
J. S. Chahal, Topics in Number Theory, Springer I. Stewart and D. Tall, Algebraic Number Theory and Fermat's Last Theorem, CRC Press J. H. Silverman, J. T. Tate, Rational Points on Elliptic Curves, Springer J.-P. Serre, A Course in Arithmetic, Springer J. Neukirch, Algebraic Number Theory, Springer J. S. Milne, Algebraic Number Theory, Course Notes F. Jarvis, Algebraic Number Theory, Springer S. Lang, Algebra, Springer H. Davenport, The Higher Arithmetic, Cambridge R. Hill, Introduction to Number Theory, World Scientific Publishing G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford
Contents
1. Elementary Number Theory: Fundamental Theorem of Arithmetic, Divisibility, prime number distribution, congruences, theorem of Fermat, Euler and Wilson, group of units, law of quadratic reciprocity, diophantine equations. 2. Algebraic Number Theory: Number fields, Dedekind rings and factorizations, class groups, quadratic fields, cyclotomic fields, Minkowski theorem, calculation of class field numbers. 3. Additional topics (depending on time, could be used for projects): glimpses into elliptic curves and rational points, Mordell's theorem
Course Language
Italian
More information
The students belonging to the categories of the project on innovative teaching will have the possibility to attend office hours also in the late afternoon and to see the notes of the lectures.
