Gallego Fernández, Alfonso2022-07-272022-07-272021-02http://hdl.handle.net/10347/28904Traballo Fin de Grao en Matemáticas. Curso 2020-2021[GL] Este traballo ten como obxectivo enunciar e demostrar o Nullstellensatz, ou Teorema dos Ceros, de Hilbert, un importante teorema alxébrico que fai de ponte entre a álxebra conmutativa e a xeometría alxébrica, permitindo deducir propiedades das variedades alxébricas afíns mediante o estudo dos ideais do anel de polinomios. Previamente enunciaranse os preliminares alxébricos e xeométricos necesarios para a realizar a proba e para comprender a súa conexión coa teoría de variedades alxébricas afíns. Logo abordarase a xeneralización do teorema a álxebras finitamente xeradas sobre aneis de Jacobson, non necesariamente sobre corpos alxebricamente pechados. Tamén enunciaranse e probaranse os casos particulares para corpos finitos e corpos reais pechados, usando as propiedades específicas que posúen cada un deses tipos de corpos.[EN] This work aims to show and prove Hilbert’s Nullstellensatz, or Theorem of Zeros, a very important algebraic theorem which acts as a bridge between commutative algebra and algebraic geometry, allowing us to deduce properties of affine varieties by studying the ideals of the polynomial ring. Previously, we will show the algebraic and geometric preliminaries required for the proof and to comprehend its connection to the theory of affine varieties. Afterwards we will show the generalization of the theorem to finitely generated algebras over Jacobson rings, not necessarily over algebraicly closed fields. We will also show and prove the particular cases of finite fields and real closed fields, using the specific properties of each of those fields.glgAtribución-NoComercial-CompartirIgual 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-sa/4.0/bachelor thesisopen access