- Docente: Lars Halvard Halle
- Crediti formativi: 6
- SSD: MAT/02
- Lingua di insegnamento: Inglese
- Modalità didattica: Convenzionale - Lezioni in presenza
- Campus: Bologna
- Corso: Laurea Magistrale in Matematica (cod. 5827)
Conoscenze e abilità da conseguire
At the end of the course, the students have an advanced knowledge in number theory and arithmetic geometry. They are able to use this knowledge both in algebraic and geometric settings.
Contenuti
Integral extensions and integral closure. Number fields and rings of integers. Norms, traces, bilinear forms and discriminants.
Dedekind domains and discrete valuation rings. Unique factorization of ideals. The ideal class group. Integral extensions and closures of Dedekind domains. Ramification theory of primes in extensions.
Class numbers of number fields. Norms of ideals. Lattices. Finiteness of the class number.
Groups of units in a number field. Finite generatedness and the rank of the group of units. CM fields and real quadratic fields.
Cyclotomic fields, their class numbers and units. Fermat's last theorem for regular primes.
Absolute values, local fields, completions. Hensel's lemma.
Algebraic curves. Rational points on curves over number fields. The Hasse principle.
Testi/Bibliografia
JAMES S. MILNE. Algebraic Number Theory.
JAMES S. MILNE. Elliptic Curves, 2006 electronic version.
Both books can be downloaded for free from author's personal webpage.
Metodi didattici
Class lectures and exercise classes (48 hours).
Modalità di verifica e valutazione dell'apprendimento
Oral exam. There is also one mandatory assignment (pass/not pass), which the student need to pass in order to be eligible for the oral exam. The grade is decided only on the basis of the oral exam.
Strumenti a supporto della didattica
Professor's lecture notes.
Orario di ricevimento
Consulta il sito web di Lars Halvard Halle