Fields, Galois theory and applications

U ovom diplomskom radu opisana je Galoisova teorija te je ista primijenjena na klasične konstrukcijske probleme u geometriji - trisekciju kuta, duplikaciju kocke, kvadraturu kruga te konstrukciju pravilnog sedmerokuta. Rad se sastoji od tri cjeline. Prvi dio služi kao podsjetnik na važne definicije i teoreme potrebne za bolje razumijevanje same Galoisove teorije. U drugoj cjelini uvodimo Galoisovu teoriju koja se razvila iz klasičnog problema teorije polinomijalnih jednadžbi. Glavna ideja Galoisove teorije je povezati proširenje polja K ⊂ F s grupom svih automorfizama od F koji fiksiraju elemente polja K, odnosno Galoisovom grupom proširenja polja K. O postojanju bijekcije između ova dva spomenuta skupa govori upravo Fundamentalni teorem Galoisove teorije. U zadnjem, trećem dijelu, ovog rada primijenili smo opisanu teoriju na konstrukcijske probleme i algebarski dokazali kako konstrukcije uistinu nisu moguće. In this thesis, Galois theory is described and applied to classical construction problems in geometry - angle trisection, doubling the cube, squaring the circle and construction of a regular heptagon. The work consists of three parts. The first part serves as a reminder of important definitions and theories necessary for a better understanding of Galois theory itself. In the second unit, we introduce the Galois theory, the origin of which begins with a classical problem in the theory of polynomial equations. The main idea of the Galois theory is to connect the field extension K ⊂ F to the group of all automorphisms of F that fix the elements of the field K, in other words, to the Galois group of the field extension K. The existence of a one-to-one correspondence between these two mentioned sets is exactly what the Fundamental theorem of Galois theory says. In the last, the third part of this paper, we applied the described theory to construction problems and proved algebraically that constructions are truly impossible.