Title Pravilni sedamnaesterokut Title (english) Regular Heptadecagon Author Krešimir Fabijančić Mentor Ivan Matić (mentor)Committee member Tomislav Marošević (predsjednik povjerenstva)Committee member Ivan Matić (član povjerenstva)Committee member Ivan Soldo (član povjerenstva)Granter Josip Juraj Strossmayer University of Osijek Defense date and country 2017-10-06, Croatia Scientific / art field, NATURAL SCIENCES Abstract Geometrijske ili euklidske konstrukcije, kako im samo ime kaže, poznate su još od
antičke Grčke. Smatralo se da se sve zna i da su iscrpljene sve mogućnosti. Međutim,
Carl Friedrich Gauss nije smatrao tako. On je uspio pokazati da uz konstrukcije poznatih
pravilnih n-terokuta kao što su trokut, četverokut i peterokut, postoje još neki
koji su konstruktibilni. Gauss je smjestio pravilne n-terokute u kompleksnu ravninu.
Vrhove pravilnih n-terokuta shvatio je kao koordinate točaka u kompleksnoj ravnini.
Obzirom da se tako svaki vrh može prikazati kao kompleksni broj z = x + yi, odnosno
u trigonometrijskom obliku z = cos ϕ + isin ϕ, za konstrukciju pravilnog n-terokuta
dovoljno je moći konstruirati dužinu duljine cos 2π/n ili sin 2π/n. To je moguće ukoliko se
riješenja jednadžbe n-tog stupnja \(x^{n}-1=0\) mogu zapisati konačnim brojem racionalnih
operacija i odredivanja drugih korijena. Rješenja takve jednadžbe nazivaju se n-ti
korijeni iz jedinice.
Koristeći se Vandermondeovim i Lagrangeovim prijašnjim rezultatima, Gauss je
vrlo vješto riješio jednadžbu sedamnaestog stupnja \(x^{17}-1=0\)
i jedno njeno rješenje
prikazao konačnim brojem racionalnih operacija i određivanja drugih korijena te tako
pokazao da je moguće konstruirati pravilni sedamnaesterokut. Rješavanje jednadžbe
\(x^{17}-1=0\) sveo je na traženje korijena ciklotomskog polinoma \(x^{16}+x^{15}+...+x+1=0\)
Ne znajući unaprijed rješenja, korijene polinoma je separirao u osam skupova po dva
korijena i tako traženje korijena polinoma šesnaestog stupnja sveo na traženje korijena
polinoma drugog stupnja. Svoje ideje i detaljne raspise postupka objavio je u knjizi
Disquisitiones Arithmeticae sa nepune 24 godine.
Općenitije, Gauss je dao točan opis brojeva n za koje je pravilni n-terokut konstruktibilan.
Ti n-ovi su usko povezani s Fermatovim brojevima i potencijama broja
dva. Formalno govoreći, Gauss je pokazao da se pravilni n-terokut može konstruirati
ravnalom i šestarom ako je n oblika \(2^{a}p_{1}p_{2}p_{3}...p_{i}\) gdje je a > 0 te \(p_{1},p_{2},p_{3},...p_{i}\)
različiti Fermatovi brojevi.
Abstract (english) Geometric or Euclidean constructions, as their name suggest, are known from antic
Greece. It was considered that all is known and all possibilities are exhausted. But,
Carl Friedrich Gauss didn’t thought so. He managed to prove that along constructions
of regular n-gons like triangle, square and pentagon, there are more of them which are
constructible. Gauss settled regular n-gons in complex plane. He imagined vertices
of regular n-gons as coordinates of points in complex plane. Considering that every
vertex can be written as complex number z = x + yi, that is in trigonometric form
z = cos ϕ+isin ϕ, for construction of regular n-gon it is enough to be able to construct
segment with length of cos 2π/n or sin 2π/n. That is possible if roots of equation of n-th
degree \(x^{n}-1=0\) can be written with finite number of rational operations and taking
of square roots. Roots of such equation are called n-th roots of unity.
Using Vandermonde and Lagrange earlier results, Gauss was very cleverly solved
equation of 17th degree \(x^{17}-1=0\)
and one of their roots wrote using finite number
of rational operations and taking of square roots and by doing so showed that it is
possible to construct regular heptadecagon. Solving of equation
he reduced \(x^{17}-1=0\)
on finding roots of cyclotomic polynomial \(x^{16}+x^{15}+...+x+1=0\)
. Not knowing
solutions in advance, he separated roots of polynomial in eight sets by two roots and
by doing so he reduced finding roots of polynomial of 16th degree into finding roots of
polynomial of 2nd degree. His ideas and detailed descriptions Gauss published in book
Disquisitiones Arithmeticae not even being 24 years old.
Generally, Gauss gave exact description of numbers n for which regular n-gon is
constructible. Those n-s are closely related to Fermat numbers and powers of number
two. Formally speaking, Gauss showed that regular n-gon can be constructed with ruler
and compass if n has form \(2^{a}p_{1}p_{2}p_{3}...p_{i}\) where a > 0 and \(p_{1},p_{2},p_{3},...p_{i}\) different
Fermat numbers.
Keywords
Gauss
Disquisitiones Arithmeticae
pravilni sedamnaesterokut
pravilni poligoni
ciklotomski polinom
korijen iz jedinice
ciklotomsko polje
Gaussov teorem
Keywords (english)
Gauss
Disquisitiones Arithmeticae
regular heptadecagon
regular polygon
cyclotomic polynomial
root of unity
cyclotomic field
Gauss theorem
Language croatian URN:NBN urn:nbn:hr:126:161274 Study programme Title: Mathematics and Computer Science Type of resource Text File origin Born digital Access conditions Open access Terms of use Created on 2017-10-23 10:20:24
