Končna polja

magistrsko delo

Alenka Vok (Author), Mateja Grašič (Mentor)

Abstract

Tema magistrskega dela je pojem, s katerim se srečujemo v algebri, to so končna polja. V delu najprej predstavimo osnovne definicije in lastnosti grup ter kolobarjev, ki jih potrebujemo za lažje razumevanje končnih polj, nato pa bolj podrobno obravnavamo polja. Polje je komutativen kolobar z enoto ▫$1\not= 0$▫ , kjer so vsi neničelni elementi obrnljivi. Vemo, da je vsako polje cel kolobar, za katerega pa velja, da ima karakteristiko enako 0 ali ▫$p$▫ , kjer je ▫$p$▫ praštevilo. Razširitev ▫$K$▫ polja ▫$F$▫ je končna, če je polje ▫$K$▫ , ki ga obravnavamo kot vektorski prostor nad poljem ▫$F$▫ , končno razsežen. Če ima končno polje ▫$F$▫ ▫$q$▫ elementov in je ▫$K$▫ končna razširitev polja ▫$F$▫, potem ima ▫$K q^n$▫ elementov, kjer je ▫$n=[K : F]$▫. Če je ▫$K$▫. razširitev polja ▫$F$▫. in ▫$f(x) \in F[x]$▫ ne konstanten polinom, ki razpade v polju ▫$K$▫ in ne razpade v nobenem pravem podpolju polja ▫$K$▫, ▫$K$▫ imenujemo razpadno polje polinoma ▫$f(x)$▫ nad ▫$F$▫. Dokažemo, da sta poljubni dve polji, ki imata končno število elementov in sta razpadni polji polinoma ▫$f(x)=x^(p^n)-x$▫ nad ▫$\mathbb{Z}_p$▫, izomorfni. Iz teh trditev sledi karakterizacija končnih polj, ki pove, da za poljubno praštevilo ▫$p$▫ in poljuben ▫$n \in \mathbb{N}$▫ obstaja do izomorfizma natančno enolično določeno končno polje s ▫$p^n$▫ elementi. Na koncu podamo enega izmed temeljnih izrekov, predstavljenih v magistrskem delu, to je Wedderburnov izrek. Izrek pove, da je vsak končen obseg polje.

Keywords

magistrska dela;karakteristike polj;karakteristike kolobarjev;klasifikacija končnih polj;razširitev polj;faktorski kolobarji;polja;končna polja;ideal;celi kolobarji;maksimalni ideal;praideal;kolobarji polinomov;homomorfizem kolobarjev;razpadna polja;vektorski prostor;Wedderburnov izrek;ničle polinomov;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[A. Vok]
UDC:	512.624(043.2)
COBISS:	23867912
Views:	1257
Downloads:	137
Average score:	0 (0 votes)
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Other data

Secondary language:	English
Secondary title:	Finite fields
Secondary abstract:	The topic, presented in this Master's thesis, are structures in the field of algebra - finite fields. We first introduce basic definitions and ring and group features to facilitate the understanding of finite fields. After that we discuss the fields. A field is a commutative ring with the unit ▫$1\not= 0$▫, where every nonzero element is inversible. Every field is an integral domain, which's characteristic is either ▫$0$▫ or ▫$p$▫ , where ▫$p$▫ is a prime number. The ▫$K$▫ extension of the field ▫$F$▫ is finite, if a field ▫$K$▫, that we consider as a vector space over the field ▫$F$▫, is finite dimensional. If the finite field ▫$F$▫ has ▫$q$▫ elements and ▫$K$▫ is the finite extension of the field ▫$F$▫, than ▫$K$▫ has ▫$q^n$▫ elements, where ▫$n =[K:F]$▫. If ▫$K$▫ is the extension of the field ▫$F$▫ and ▫$f(x) \in F[x]$▫ is a nonconstant polynomial, that splits in the field ▫$K$▫ and doesn't split in none of the proper subfields of field ▫$K$▫, than ▫$K$▫ is the splitting field of the polynomial ▫$f(x)$▫ over ▫$F$▫. We prove, that any two fields with finite number of elements that are also splitting fields of the polynomial ▫$f(x)=x^(p^n)-x$▫ over ▫$\mathbb{Z}_p$▫,, are isomorphic. Complying with those claims the characterization of the finite fields follows, which states, that for any prime number ▫$p$▫ and any positive integer ▫$n$▫, there is, up to isomorphism, a unique finite field of ▫$p^n$▫ elements. In the end we discuss one of the basic theorems in this thesis, the Wedderburn's theorem, which states that every finite division ring is a field.
Secondary keywords:	master theses;characteristics of fields;characteristics of rings;classification on finite fields;extension fields;factor rings;fields;ideal;ideal domain;maximal ideal;prime ideal;polynomial rings;rings homomorphism;splitting fields;vector space;Wedderburn's theorem;zeros of polynomials;
URN:	URN:SI:UM:
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	XI, 78 f.
ID:	10901609