| Abstract | U prvom smo poglavlju proučili općenito grupe i prstene, te smo definirali što je to potpuno uređeno polje. U drugom smo se poglavlju osvrnuli na prirodne, cijele i racionalne brojeve, te ih definirali unutar fiksiranog polja realnih brojeva. Zatim smo u trećem poglavlju proučili potprstene i potpolja. U četvrtom poglavlju smo definirali polje kompleksnih brojeva \(\mathbb{C}\), nakon čega smo proučavali određeno potpolje od \(\mathbb{C}\) izomorfno s \(\mathbb{R}\). Pri kraju poglavlja smo definirali morfizam uređenih skupova. U petom poglavlju smo proučavali svojstva tijela kvaterniona \(\mathbb{H}\), morfizme i izomorfizme prstenova. Također, razmotrili smo određeno polje u \(\mathbb{H}\) koje je izomorfno sa \(\mathbb{C}\). Na kraju rada, u šestom poglavlju smo definirali prsten hiperboličkih brojeva \(\mathbb{D}\) i proučili određeno polje u \(\mathbb{D}\) koje je izomorfno sa \(\mathbb{R}\). |
| Abstract (english) | In the first chapter, we studied groups and rings in general, and defined what a totally ordered field is. In the second chapter, we looked at natural, integer, and rational numbers, and defined them within a fixed field of real numbers. Then, in the third chapter, we studied subrings and subfields. In the fourth chapter, we defined a field of complex numbers \(\mathbb{C}\), after which we studied a certain subfield of \(\mathbb{C}\) isomorphic to \(\mathbb{R}\). At the end of the chapter, we defined morphism of ordered sets. In the fifth chapter, we studied the properties of a field of quaternions \(\mathbb{H}\), morphisms, and ring isomorphisms. We also considered a particular field in \(\mathbb{H}\) that is isomorphic to \(\mathbb{C}\). At the end of the paper, in sixth chapter, we defined a ring of hyperbolic numbers \(\mathbb{D}\) and studied a particular field in \(\mathbb{D}\) that is isomorphic to \(\mathbb{R}\). |
