Algebraic Number Theory

A.Y. 2019/2020
Overall hours
Learning objectives
The course provides standard results in algebraic number theory, formulated in the classical language.
Expected learning outcomes
Learning the basic results in Algebraic Number Theory. Ability of computing the class groups and the group of units of a number field.
Course syllabus and organization

Single session

Lesson period
Second semester
Prerequisites for admission
Basic knowledge of algebra (Algebra 1-4) and analysis (Analisi Matematica 1-4).
Assessment methods and Criteria
The exam consists of a written test and an oral examination in the same session. The written part is made of written exercises (like a computation of the ring of
integers, the class group and the group units of a number field). It is not
possible to use notes, books or calculators.
Number Theory (first part)
Course syllabus
First properties of a number field: norm, trace, discriminant and ring of integers (review of some of the arguments of the Algebra 3 course). Dedekind rings, factorization of ideals and ramification. Theorem of Minkowski. Theorem of Hermite. Theorem of Dirichlet and regulator of a number field. Dedekind ζ function. Class number formula. Other more advanced topics, time permitting.
Teaching methods
Whole class teaching of theory and excercises.
Teaching Resources
R. Schoof's notes, "Algebraic number theory" available at
Number Theory mod/2
Course syllabus
L-functions, density theorems and reciprocity laws.
Teaching methods
Whole class teaching of theory and excercises.
Teaching Resources
J. W. S. Cassels and A. Frohlich (editors), "Algebraic number theory".
Number Theory (first part)
MAT/02 - ALGEBRA - University credits: 6
Lessons: 42 hours
Professor: Seveso Marco Adamo
Number Theory mod/2
MAT/02 - ALGEBRA - University credits: 3
Lessons: 21 hours
On appointment
Via Cesare Saldini 50