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

[2] Fesenko I. - Core Topics in Number Theory I
[3] Boucksom S. - Successive minima and lattice points

Exercise sheets
Sheet 1         Final Exam 23/12/2024

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Classroom diary (September-December 2024):
1 - 03/09 - Integral elements. Integral closure. Ring of integers.  Ref. [2, Sect. 2.1].

2 - 10/09 - I was away. I. Fesenko covered for me
3 - 14/09 - I was away. I. Fesenko covered for me
4 - 24/09 - Dedekind domains and their basic properties: cancellation property; factorization property; unique factorization of ideals; any ideal is generated by at most two elements Ref. [2, Sect. 3.3].
5 - 08/10 - Fractional ideals. Unique factorization of fractional ideals. The class group. Theory of lattices. Characterization of lattices as discrete subgroups. Characterization of complete lattices. Minkowski's first theorem. Blichtfeld's principle.  Ref. [1, I.4], [3].
6 - 15/10 - Minkowski vector space. Ideals as complete lattices. Theorem needed for the finiteness of the class number. Multiplicative Minkowski theory.  Ref. [1, I.5].​

7 - 22/10 - Norm of an ideal of the ring of integers. The finiteness of the class number. The fundamental exact sequence involving the units of the ring of integers. Dirichlet unit theorem.  Ref. [1, I.6 - I.7].​​

8 - 29/10 - Further discussion around the Dirichlet unit theorem. Principal units and the regulator of a number field. Extension of Dedekind domains. Ramification index and inertia degree. Geometric picture of the ramification phenomenon. Ref. [1, I.7 - I.8].​

9 - 05/11 - Proof of the theorem that relates the ramification indexes and the inertia degrees. Extension of prime ideals: explicit factorization for almost all primes. Only finitely many primes can ramify. Hilbert's ramification theory: action of the Galois group on primes; Galois action in the case of a Galois field extension. Ref. [1, I.8 - I.9].

10 - 12/11 - Decomposition group and decomposition field. Factorization of primes for Galois extensions. Inertia group and inertia field.  Ref. [1, I.9].

11 - 19/11 - DVR and their characterizations. Relation between DVR and localizations of Dedekind domains. Absolute values on fields. Ref. [1, I.11, II.3].

12- 26/11 - Problem session
13- 03/12 - Characterization of equivalent absolute values. Weak approximation theorem. Ostrowski lemma. Completions.  Ref. [1, II.3, II.4].
14- 10/12 - Elements of complete DVF expressed as power series. Completion of DVF as projective limit. Hensel lemma and some consequences. Ref. [1, II.4].
15- 17/12 - Ostrowski theorem. Algebraic extension of complete valued fields. Local fields and their characterization. The case of local DVF.  Ref. [1, II.5].