The course provides standard results in algebraic number theory, hence introduce L-functions and their arithmetic relevance.
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. Acquire familiarity with L-functions and other more advanced topics.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
Basic knowledge of algebra (Algebra 1-4) and analysis (Analisi Matematica 1-4).
Assessment methods and Criteria
L'esame consiste di una discussione orale.
Number Theory (first part)
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.