2013/2014

2014/2015

DM270

MAT/02 (ALGEBRA)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

1st semester (01/10/2014 - 15/01/2015)

6

48 lesson hours

ITALIAN

ORAL TEST

PERNAZZA LUDOVICO (titolare) - 3 ECTS
CANONACO ALBERTO - 3 ECTS

The contents of the courses of Algebra 1, Algebra 2 and Linear Algebra

An introduction to the fundamental notions of Commutative Algebra and Algebraic Number Theory.

Review of previously known results on commutative rings and ideals; modules and operations on them; tensor product of modules. Localization of rings and modules.
Artinian and Noetherian rings and modules; Krull dimension of a ring; length of a module.
Integral dependence; integral extensions; integral closure of a domain; valuation rings; discrete valuation rings; Hilbert's Nullstellensatz.
Dedekind domains: fractional ideals, unique factorization of ideals; ideal class group.
Number fields: ring of integers; norm, trace and discriminant; quadratic reciprocity.
Lattices, Minkowski's lemma and Lagrange's four-squares theorem.
Finiteness of the class number and unit theorem.

Lectures

M.F. Atiyah, I.G. MacDonald: "Introduction To Commutative Algebra", Addison-Wesley, 1994.
G.J. Janusz: "Algebraic Number Fields", American Mathematical Society, 2005.
I. Kaplanski: "Commutative Rings", University of Chicago Press, 1974.
S. Lang: "Algebraic Number Theory", Springer, 1994.
H. Matsumura: "Commutative Ring Theory", Cambridge Univerity Press, 1989.
J.S. Milne: "Algebraic Number Theory", 2012.

Oral exam

Oral exam