Hilbert's Nulls Theorem

Teme istražene u ovom radu spadaju u domenu komutativne algebre, obuhvaćajući i aspekte homološke algebre. Komutativna algebra čvrsto je povezana s algebarskom geometrijom, što predstavlja motivaciju za razvoj određenih koncepata unutar komutativne algebre. Osim toga, ovo je područje gdje se rezultati dobiveni proučavanjem tih koncepata mogu primijeniti u praksi. Iako su odnosi između geometrijskih objekata i matematičkih jednadžbi koje ih opisuju proučavani tijekom stoljeća, specifična povezanost između ovih dvaju područja nije bila potpuno jasna sve do sredine 19. stoljeća. U tom vremenskom razdoblju, David Hilbert je igrao ključnu ulogu svojim rezultatima, uključujući teorem o bazi, teorem o nulama, koncept polinomijalne strukture poznate kao Hilbertova funkcija i teorem o sizigiji. Ovi rezultati su postavili temelje komutativne algebre. Ovaj rad započinje analizom afinih algebarskih skupova, a zatim se bavi idealima, s posebnim fokusom na koordinatne prstene, k-algebre konačnog tipa i maksimalne ideale. Završno poglavlje istražuje temu "malog" i "velikog" Hilbertovog teorema o nulama te se bavi njihovim aspektima.The topics covered in this text belong to the commutative algebra domain, including homological algebra elements. Commutative algebra is closely linked to algebraic geometry, which serves as motivation for the development of certain concepts within commutative algebra.Moreover, this is an area where results obtained by studying these concepts can be practically applied. Although the relationship between geometric objects and the mathematical equations that describe them has been studied for centuries, the specific connection between these two fields discussed here was not fully recognized until the mid-nineteenth century. During that period, David Hilbert’s results, including the basis theorem, the nullstellensatz, the polynomial structure known as Hilbert’s function, and the syzygy theorem, played a crucial role in laying the foundations of commutative algebra. This paper begins with an analysis of affine algebraic sets, followed by a discussion of ideals, with a particular focus on coordinate rings, finite-type k-algebras, and maximal ideals. The final chapter explores the topic of the "small" and "big" Hilbert nullstellensatz theorems and delves into their aspects