Algebraic Number Theory
A.Y. 2023/2024
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: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Algebraic number theory (activated first part)
Responsible
Lesson period
Second 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)