Abel-Ruffinijev teorem

master's thesis

• Author: Josipa Pavković
• Mentor: Ozren Perše (mentor)
• Committee member: Ozren Perše (predsjednik povjerenstva)
• Committee member: Zlatko Drmač (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-02-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Tema ovog diplomskog rada je Abel-Ruffinijev teorem koji nam govori da ne postoji opće algebarsko rješenje, tj. rješenje u radikalima za jednadžbe petog ili višeg stupnja. U prvom poglavlju uvodimo neke osnovne definicije potrebne za izgradnju Galoisove teorije, kojom se opširnije bavimo u četvrtom poglavlju. U drugom poglavlju poseban naglasak stavit ćemo na prsten polinoma. Vidjeti ćemo da, kada je k polje, svi poznati teoremi koji vrijede u Z, imaju anologon u k[x], štoviše, vidjeti ćemo da se svi poznati dokazi mogu prenijeti ovdje. U trećem poglavlju definiramo kvocijentni prsten i uvodimo pojam proširenja polja. U petom poglavlju ukratko ćemo pokazati postupak rješavanja jednadžbi trećeg i četvrtog stupnja. U zadnjem poglavlju dokazuje se Galoisov teorem o korespondenciji rješivosti polinoma u radikalima i rješivosti pripadne Galoisove grupe, te Abel-Ruffinijev teorem.

Abstract (english)

The theme of this diploma thesis is Abel-Ruffini theorem which states that there is no general algebraic solution, that is, solution in radicals, to polynomial equations of degree five or higher. In Chapter 1 we introduce some basic definitions that will be needed to build the foundations of the Galois theory, which we detail in Chapter 4. In Chapter 2 we give a special attention to the ring of polynomials. We are going to see that, when k is a field, virtually all the familiar theorems valid in Z have polynomial analogs in k[x]; moreover, we shall see that the familiar proofs can be generalized. In the third chapter we define quotient ring and introduce the notion of field extension. In Chapter 5 we will briefly show the process of solving equations of the third and fourth degree. In the last chapter we will prove Galois theorem on correspondence between solvability of polynomial in radicals and the corresponding Galois group, and Abel-Ruffini theorem.

• Keywords: Abel-Ruffinijev teorem; Galoisova teorija; prsten polinoma; Galoisov teorem o korespondenciji rješivosti polinoma u radikalima; Galoisova grupa
• Keywords (english): Abel-Ruffini theorem; Galois theory; ring of polynomials; Galois theorem on correspondence between solvability of polynomial in radicals; Galois group
• Language: croatian
• URN:NBN: urn:nbn:hr:217:400915
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-29 09:55:23