ALGEBRA III

Course objectives

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

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PAOLO PAPI Lecturers' profile

Program - Frequency - Exams

Course program
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.
Prerequisites
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.
Books
J.S. Milne, Algebraic number theory M. Atiyah, I Macdonald, Introduction to commutative algebra R.B. Ash A Course in Algebraic number theory
Frequency
Recommended
Exam mode
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.
Bibliography
Neukirch, Algebraic number theory
Lesson mode
Frontal lessons. A third of the lessons will be dedicated to complements and exercises
  • Lesson code1022448
  • Academic year2025/2026
  • CourseMathematics
  • CurriculumGenerale
  • Year3rd year
  • Semester2nd semester
  • SSDMAT/02
  • CFU6