Guitart Morales, XavierGil Muñoz, Daniel2018-03-272018-03-272017-09-10https://hdl.handle.net/2445/121135Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2017, Director: Xavier Guitart Morales[en] We present and describe a proof due to Deuring of the Hilbert’s 12th problem in the particular case of imaginary quadratic fields. This particular case consists in giving explicit generators of the maximal abelian extension of an imaginary quadratic field. To deal with abelian extensions of such fields, we study the classical formulation of Class Field Theory and present the main statements without proofs. The other important ingredient for the proof we will describe is the theory of elliptic curves with complex multiplication. We will combine Class Field Theory with the notions of elliptic curves to prove the desired construction.96 p.application/pdfengcc-by-nc-nd (c) Daniel Gil Muñoz, 2017http://creativecommons.org/licenses/by-nc-nd/3.0/es/Corbes el·líptiquesTeoria de cossos de classeTreballs de fi de màsterExtensions abelianesTeoria de torsió (Àlgebra)Geometria algebraica aritmèticaVarietats abelianesElliptic curvesClass field theoryMaster's thesesAbelian extensionsTorsion theory (Algebra)Arithmetical algebraic geometryAbelian varietiesinfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/openAccess