Mathematics

From the journal of the course, year 2024-2025 General facts on Hom functors and tensor products. Flat modules. Tensor products in free resolutions. Projective modules. Locally free modules. Background on Dedekind rings. Various equivalent conditions. Fractional and invertible ideals. Every invertible fractional ideal is projective. Prime ideals inert, decomposed, ramified. Notion of lattice index (Fröhlich). Theorem: The index of two A-lattices is a non-zero fractional ideal. AKLBG formalism. Action of the Galois group on the group of fractional ideals. Transitivity of the action on the primes that divide a given prime. Example: Artin-Schreier extensions. ramification. Recall on the fundamental theorem of Galois theory (finite extensions). interpretation of the formula [L:K]=efg, the symbol of Artin. Definition and first properties. Example: Quadratic extensions of Q: Kronecker symbol. Formalism AKLB in case A is complete DVR. Example: p-adic integers. Hensel's lemmas. Lying-over-Theorem LOT, Going up, Dimension in integral extensions. Dedekind-Kummer theorem and local version Totally ramified extensions: (local) equivalence criterion, Eisenstein polynomials. Explicit characterization of totally tamely ramified local extensions. Concrete categories with products. Equalizers. Projective limit, universal property of the projective limit (without proofs). Projective limits of Hausdorff topological spaces, groups, rings. Consequences of Tychonov's theorem. Equivalent norms, weak approximation theorem. Structure of the completion of a DVR, interpretation as a projective limit. Structure arborescent of an open system. Consequence: Hausdorff totally disconnected and compact if and only if finite residual field. Definition of local field. Examples: Ring of pro finite integers. Unramified extensions (complete local case). maximal unramified sub-extension. Existence of category equivalence between unramified and separable extensions of residual field. Consequences of the category equivalence for unramified extensions (local case). Finite étale algebras and fundamental properties. Basic extensions for finite étale algebras. Structure theorem for completions in the AKLB case. Calculation of decomposition and inertia groups. Kronecker-Weber theorem. Main ingredients, statement of the global and local case. Kronecker Weber theorem: the local case implies the global one. Proof of the local Kronecker-Weber theorem in p, case of cyclic extensions of degree l^k with l different from p. Notes on the case l=p odd (Kummer theory is used), and on the case l=p=2." "Maximal Abelian extension of Qp. Decomposition of the Galois group. Structure of Qp^x Local reciprocity law for Qp. Infinite Galois correspondence (Krull). Examples. Account on Artin's global reciprocity.

It is quite recommended to have taken the Algebra 3 course. Basics of commutative algebra (see Atiyah McDonald's book) are used without reminders. Rudiments of Galois theory are also used, in conjunction with the theory of metric and topological spaces.

Atiyah McDonald Introduction to Commutative Algebra (background) Eisenbud Commutative Algebra with a View Toward Algebraic Geometry Koblitz p-adic Numbers, p-adic Analysis, and Zeta-Functions Neukirch Algebraic Number Theory Sutherland Number Theory (MIT OpenCourseWare) Washington Introduction to Cyclotomic Fields

'Frontal teaching'. More info available in the department's website.

To be decided later: oral exam or project or both

Atiyah McDonald Introduction to Commutative Algebra (background) Eisenbud Commutative Algebra with a View Toward Algebraic Geometry Koblitz p-adic Numbers, p-adic Analysis, and Zeta-Functions Neukirch Algebraic Number Theory Sutherland Number Theory (MIT OpenCourseWare) Washington Introduction to Cyclotomic Fields

Traditional lessons on the blackboard. See website if the department Guido Castelnuovo.