Gauss-Wantzelov izrek

magistrsko delo

Luka Grahelj (Author), Aleš Vavpetič (Mentor)

Abstract

V magistrskem delu predstavimo Gauss-Wantzelov izrek, ki nam pove, za katera naravna števila $n$ je pravilne $n$-kotnike mogoče konstruirati zgolj z rabo ravnila in šestila. Izrek to lastnost števil najprej analizira na njihovih praštevilskih (oznaka $p$) gradnikih, za katere ugotavlja, da so pravilni $p$-kotniki konstruktibilni natanko tedaj, ko so praštevila $p$ Fermatova (oznaka $p_F$). Pri tem je vsak izmed avtorjev izreka prispeval dokaz ene smeri te ekvivalence: Gauss je najprej našel algoritem, s katerim lahko za poljubno Fermatovo praštevilo $p_F$ naposled vedno skonstruiramo ustrezni $p_F$-kotnik, Wantzel pa je dokazal, da niti teoretično ne bi bilo mogoče skonstruirati drugačnih $p$-kotnikov kot prav tistih, za katere je to storil že Gauss. Središčne kote med seboj različnih $p_{F_i}$-kotnikov lahko z ustreznimi celoštevilskimi kombinacijami seštejemo v središčni kot $\prod_i p_{F_i}$-kotnika, ob tem pa ga lahko z zaporednimi bisekcijami še poljubnokrat razpolovimo, v čemer imamo tako tudi algoritem, kako konstruirati pravilne večkotnike za sestavljena števila $s$ v obliki $s=2^kp_{F_1}p_{F_2}\ldots p_{F_t}$. Kot pri konstruiranju posameznih $p$-kotnikov se tudi pri njihovem sestavljanju v $s$-kotnike izkaže, da se zadostni pogoj za uporabo tega algoritma samega že pokriva s potrebnim, zato so le-ti konstruktibilni za natanko opisane oblike števil $s$.

Keywords

konstrukcije z ravnilom in šestilom;pravilni večkotniki;koreni enote;ciklotomični polinomi;Gaussove periode;Wantzelov sistem;Fermatova praštevila;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[L. Grahelj]
UDC:	512.62
COBISS:	18365017
Views:	1061
Downloads:	431
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The Gauss-Wantzel Theorem
Secondary abstract:	The subject of this paper is the Gauss-Wantzel Theorem that states which regular $n$-gons can be constructed using only straightedge and compass. Said property of natural numbers $n$ is first analyzed among their basic building blocks in the form of primes (denoted by $p$), for which the theorem determines that regular $p$-gon is constructible if and only if $p$ is a Fermat prime (denoted by $p_F$). In that regard, each of the authors provided the proof of one of the directions of the proposed equivalence: Gauss first developed an algorithm that allows us to eventually construct a regular $p_F$-gon for any Fermat prime $p_F$, whereas Wantzel proved that no regular $n$-gons others than the ones already suggested by Gauss could ever be constructed. Using appropriate integer combinations, central angles of different $p_{F_i}$-gons can be added into central angle of a $\prod_i p_{F_i}$-gon, which can further be repeatedly divided into two by consecutive angle bisections. Hence we have an algorithm on how to construct a regular $c$-gon for composite numbers $c$ in the form $c=2^kp_{F_1}p_{F_2}\ldots p_{F_t}$. As with the construction of single $p$-gons, it turns out that the sufficient condition for the application of this particular algorithm for their composition already aligns with the necessary one, therefore making the aforementioned $c$-gons precisely the ones being constructible.
Secondary keywords:	compass-and-strightedge constructions;regular polygons;roots of unity;cyclotomic polynomials;Gaussian periods;Wantzel system;Fermat primes;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Pedagoška matematika
Pages:	X, 55 str.
ID:	10933760