|
|
Gauss and the constructability of regular polygons with ruler and compass
Sedláčková, Veronika ; Kvasz, Ladislav (advisor) ; Jančařík, Antonín (referee)
1 ABSTRACT The bachelor thesis deals with chosen Euclidean constructions of regular polygons and summarizes their historical development. It focuses on the mathematician who is essentially adherent to this theme, his name is Carl Friedrich Gauss. In the first part of the thesis the important statements from Gauss's life and particularly from his scientific publications are given. Then the idea of algebraic formulation of the constructions with ruler and compass is characterized and the main theorems about these constructions are proved here. Further Gauss's theorem about constructability of regular polygons is given and proved while using Galois Theory. The next part is focused on Gauss's construction of the regular 17-gon, which is described in details. At the same time the thesis explains other interesting constructions from various authors created during the 19th century and in the beginning of the 20th century.
|
|
|
Évariste Galois and his theory
Richter, Lukáš ; Kvasz, Ladislav (advisor) ; Jančařík, Antonín (referee)
TITLE: Évariste Galois and His Theory AUTHOR: Lukáš Richter DEPARTMENT: The Department of Mathematics and Teaching of Mathematics SUPERVISOR: prof. RNDr. Ladislav Kvasz, Dr. ABSTRACT: The first part of my thesis deals with the life of the significant French mathematician of the 19th century, Évariste Galois, the founder of modern algebra. It is focused on his school years, meetings with mathematics, unsuccessful entrance exams to university, expulsion from school, his mathematic works and his bad experiences with French Academy of Science. He had difficulties with law during his life twice, he was judged and imprisoned. At the end of his short life he fell in love unhappily and consequently was killed in a duel. The second part is devoted to the solution of polynomial equations of the first degree up to the fourth degree by the algebraic patters already known at the times of Galois. Each of the formula is derived by the method suitable even for the students of secondary schools and its usage is illustrated on the example. The third part contains the basics of the Galois Theory and the insolubility of polynomial equations of at least fifth degree is demonstrated. Some of the statements are introduced on examples. KEYWORDS: Évariste Galois, polynomial equations, field extensions, automorphism groups, Galois Theory
|
|
|
Gauss and the constructability of regular polygons with ruler and compass
Sedláčková, Veronika ; Kvasz, Ladislav (advisor) ; Jančařík, Antonín (referee)
1 ABSTRACT The bachelor thesis deals with chosen Euclidean constructions of regular polygons and summarizes their historical development. It focuses on the mathematician who is essentially adherent to this theme, his name is Carl Friedrich Gauss. In the first part of the thesis the important statements from Gauss's life and particularly from his scientific publications are given. Then the idea of algebraic formulation of the constructions with ruler and compass is characterized and the main theorems about these constructions are proved here. Further Gauss's theorem about constructability of regular polygons is given and proved while using Galois Theory. The next part is focused on Gauss's construction of the regular 17-gon, which is described in details. At the same time the thesis explains other interesting constructions from various authors created during the 19th century and in the beginning of the 20th century.
|
|
|
Why quintic polynomial equations are not solvable in radicals
Křížek, Michal ; Somer, L.
We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed bz radicals, i.e., by the operations +, -, ., :, and .... Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.
|
guest ::
