Mathematics

PAOLO PAPI

General objectives: acquire knowledge in commutative algebra and algebraic number theory
Specific objectives
Knowledge and understanding
At the end of the course students will have acquired the main basic notions of Commutative Algebra,
concerning commutative rings with or without zero dividers, whole ring extensions, tensor products e
flatness, rings and Artinian and Noetherian modules, including the dimension theory for k-algebra
finitely generated, primary decomposition, Dedekind domains, ramifications, class number

Ability to apply knowledge and understanding
At the end of the course students will be able to apply the knowledge acquired competently
and thoughtful and solve simple problems that require the use of techniques related to Commutative Algebra and algebraic theory
of numbers.

Judgment autonomy
The student will have the basis for analyzing the analogies and relationships between the topics covered and
topics of Geometry or Algebraic Number Theory and will have an idea of ​​how important these are
branches of Mathematics are also historically deeply linked.

Communication skills
• The student will be able to present the main theorems with their proofs in the context of
oral test and will be able to communicate the key ideas of Algebra to non-specialists
Commutative and algebraic number theory.
.

Learning ability
The student will be able to put to use the topics of Commutative Algebra and algebraic theory of numbers learned in the numerous
mathematical contexts in which they are used, both in the context of the Master's degree courses, and in
a future research activity

The course is an introduction to commutative algebra and algebraic number theory.
At the end of the course the students will have acquired the main basic notions of Commutative Algebra, concerning commutative rings with or without zero divisors, integer extensions of rings, Artinian and Noetherian rings and modules, primary decomposition, Dedekind domains, ramifications, class number

Good knowledge of elementary algebra that can be acquired in the Algebra course: groups, rings, fields. Good knowledge of linear algebra, which can be acquired in the Algebra Linere course. It is desirable to know the Galois correspondence, at least for finite extensions in characteristic zero. There is no preliminary exam formally needed to take the course.

Review of ring theory; nil radical; local rings
quotient ideals , radical, localization
A-modules: general notions on A-modules; Nakayama lemma
prime avoidance Lemma, conditions on chains: Artinian and Noetherian modules
Noetherian rings: Hilbert basis theorem, properties and primary decomposition of Noetherian rings
Artinian rings
Integral rings
Recollections on norms and traces
Discrete valuation rings and Dedekind domains
AKLB setup
discriminating
Fractional ideals
Factorization of ideals in Dedekind domains
Ramification, residual degree, fundamental relationship
Dedekind's theorem
Factorization in quadratic fields, canonical embedding, Minkowski's theorem, finiteness of the class group
Examples of use of the Minkowski bound; preparation for Dirichlet's theorem
Dirichlet theorem on the structure of the group of invertibles in the ring of integers of a number field; quadratic reciprocity
AKLB setup in Galoisian extensions, Frobenius automorphism, proof of the quadratic reciprocity theorem.

J.S. Milne, Algebraic number theory
M. Atiyah, I Macdonald, Introduction to commutative algebra
R.B. Ash A Course in Algebraic number theory

Neukirch, Algebraic number theory

Frontal lessons. A third of the lessons will be dedicated to complements and exercises

Recommended

The exam aims to evaluate learning through a written test (consisting in solving problems of the same type as those carried out in the exercises) and an oral test (consisting in the discussion of the most relevant topics illustrated in the course). The written test will last approximately three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place in the middle of the course and the second immediately at the end of the course.
To pass the exam it is necessary to achieve a grade of not less than 18/30. The student must demonstrate that he has acquired sufficient knowledge of the subjects of commutative algebra and number theory, and that he is able to perform at least the simplest of the assigned exercises.
To achieve a score of 30/30 cum laude, the student must instead demonstrate that he has acquired an excellent knowledge of all the topics covered during the course and be able to connect them in a logical and coherent way.

It is not appropriate to provide standard questions for courses of this type: it will still be necessary to know all the definitions introduced in the course, accompanied by examples and non-examples and the formulation of the main theorems