Galoisova teorija

master's thesis

• Author: Dragutina Topić
• Mentor: Ozren Perše (mentor)
• Committee member: Ozren Perše (predsjednik povjerenstva)
• Committee member: Zvonimir Tutek (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-11-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad smo podijelili u dva dijela. U prvom dijelu smo dali kratki podsjetnik na teoriju grupa, Lagrangeov teorem, kvocijentne skupove, te komutativne prstene, polja razlomaka, te polinome. Uvodni dio nam je bio od velike važnosti kasnije za shvaćanje same Galoisove teorije. U drugom dijelu smo se upoznali s Galoisovom teorijom, čiji je originalni naziv teorija jednadžbi. Odmah na početku smo definirali Galoisovu grupu. Sama Galoisova teorija nam daje vezu između algebarskog proširenja \(E\) nad poljem \(k\), te pripadne Galoisove grupe \(Gal(E/k)\). Naveli smo nekoliko teorema o Galoisovim grupama čije tvrdnje kažu da su proširenja polja također i polja razlaganja nekog polinoma. Na samom kraju smo iskazali i dokazali Fundamentalni teorem Galoisove teorije, kao posljedicu cjelokupne analize drugog dijela.

Abstract (english)

This thesis is divided into two parts. The first part was a brief reminder of the group theory, Lagrange’s theorem, quotient groups, and commutative rings, fields fractions, and polynomials. The introductory part was of great importance for the understanding of Galois theory in the second part. In the second part, we got familiar with Galois theory, which original name was the theory of equations. Right at the beginning we defined the Galois group. Galois theory gives connection between algebraic extension \(E\) over the field \(k\), and the corresponding Galois group \(Gal(E/k)\). We stated some theorems about Galois groups which say that an extension is a splitting field of a polynomial. At the very end we stated and proved the fundamental theorem of Galois theory, as a result of the overall analysis of the second part.

• Keywords: Galoisova teorija; Galoisova grupa; Fundamentalni teorem Galoisove teorije
• Keywords (english): Galois theory; Galois group; fundamental theorem of Galois theory
• Language: croatian
• URN:NBN: urn:nbn:hr:217:872806
• 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: 2017-03-29 12:53:12