References:
[1] Neukirch J. - Algebraic Number theory

[2] Fesenko I. - Core Topics in Number Theory I


 

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Classroom diary (September-December 2025):
1 - 08/09 - Integral elements. Integral closure. Ring of integers. Norm and Trace. Discriminants. Ref. [1, I.2].

2 - 15/09 - Norm and trace with respect to integral structures. Integral bases. Discriminant of a number field. Dedekind domains. Ref. [1, I.2, I.3].

3 - 22/09 - Unique factorization of ideals in Dedekind domains. The group of fractional ideals and the class group. Ref. [1, I.3].

4 - 29/09 - Lattice theory. Minkowski first theorem. Ideals seen as lattices in the Minkowski space of a number field. Ref. [1, I.4, I.5].

5 - 13/10 - Theorem needed for the finiteness of the class number. Multiplicative Minkowski theory. Norm of an ideal of the ring of integers. Ref. [1, I.5, I.6]

6 - 20/10 - Finiteness of the class number. Dirichlet unit theorem. Regulator of a number field. Ref. [1, I.6, I.7]

7 - 27/10 - Extensions of Dedekind domains and factorization of extended primes. Ramification, inertia and the fundamental formula. Conductor. Ref. [1, I.8]

8 - 03/11 - Dedekind-Kummer theorem on extension of Dedekind domains. Finiteness of ramified primes. Hilbert ramification theory (part I). Ref. [1, I.8, I.9]

9 - 10/11 - Hilbert ramification theory (part II). Ring of integers of cyclotomic extensions. Ref. [1, I.9, I.10]

10 - 17/11 - Factorization of primes in cyclotomic extensions. Quadratic reciprocity law. Discrete valuation rings. Ref. [1, I.10, I.11]

11 - 24/11 - Localizations of Dedekind domains and discrete valuations. Dirichlet's theorem  for S-units. Absolute values. Ref. [1, I.11, II.3]