Algebraic Number Theory
A.Y. 2024/2025
Learning objectives
(first part) The course provides standard results in algebraic number theory, hence introduce L-functions and their arithmetic relevance.
Expected learning outcomes
(first part) 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: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Algebraic number theory (activated first part)
Responsible
Lesson period
First semester
Prerequisites for admission
Basic knowledge of algebra (Algebra 1-4) and analysis (Analisi Matematica
1-4 and Analisi complessa).
1-4 and Analisi complessa).
Assessment methods and Criteria
The exam consists of an oral examination.
Number Theory (first part)
Course syllabus
First properties of a number field, review of some of the arguments of
the Algebra 3 course (Dedekind rings, factorization of ideals and ramication).
Theorem of Minkowski. Theorem of Hermite. Theorem of Dirichlet and regulator
of a number field. Dedekind Zeta function. Class number formula. Complex
L functions, special value formulas and relationship with cyclotomic units.
the Algebra 3 course (Dedekind rings, factorization of ideals and ramication).
Theorem of Minkowski. Theorem of Hermite. Theorem of Dirichlet and regulator
of a number field. Dedekind Zeta function. Class number formula. Complex
L functions, special value formulas and relationship with cyclotomic units.
Teaching methods
Whole class teaching of theory and excercises.
Teaching Resources
-J. W. S. Cassels and A. Frohlich, "Algebraic number theory", Academic Press Inc. (London) LDT, 1967.
-S. Lang, "Algebraic number theory", Springer, 1994.
-S. Lang, "Cyclotomic fields I and II", combined 2nd edition, Springer, 2012.
-D. A. Marcus, "Number fields", Springer, 2018 (2nd edition).
-R. Schoof, "Algebraic number theory". Notes avalable at https://www.mat.uniroma2.it/~eal/moonen.pdf.
-J.-P. Serre, "Local fields", Springer.
-L. C. Washington, "Introduction to cyclotomic fields", 2nd edition, Springer,
1997.
-S. Lang, "Algebraic number theory", Springer, 1994.
-S. Lang, "Cyclotomic fields I and II", combined 2nd edition, Springer, 2012.
-D. A. Marcus, "Number fields", Springer, 2018 (2nd edition).
-R. Schoof, "Algebraic number theory". Notes avalable at https://www.mat.uniroma2.it/~eal/moonen.pdf.
-J.-P. Serre, "Local fields", Springer.
-L. C. Washington, "Introduction to cyclotomic fields", 2nd edition, Springer,
1997.
Number Theory mod/2
Course syllabus
Other arithmetic applications of L-functions.
Teaching methods
Whole class teaching of theory and excercises.
Teaching Resources
-J. W. S. Cassels e A. Frohlich, "Algebraic number theory", Academic Press Inc. (London) LDT, 1967.
-S. Lang, "Algebraic number theory", Springer, 1994.
-S. Lang, "Cyclotomic fields I and II", combined 2nd edition, Springer, 2012.
-J.-P. Serre, "Local fields", Springer.
-L. C.Washington, "Introduction to cyclotomic fields", 2nd edition, Springer,
1997.
-S. Lang, "Algebraic number theory", Springer, 1994.
-S. Lang, "Cyclotomic fields I and II", combined 2nd edition, Springer, 2012.
-J.-P. Serre, "Local fields", Springer.
-L. C.Washington, "Introduction to cyclotomic fields", 2nd edition, Springer,
1997.
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
Professor(s)