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Title:Končna polja
Authors:Vok, Alenka (Author)
Grašič, Mateja (Mentor) More about this mentor... New window
Files:.pdf MAG_Vok_Alenka_2018.pdf (661,24 KB)
 
Language:Slovenian
Work type:Master's thesis/paper (mb22)
Organization:FNM - Faculty of Natural Sciences and Mathematics
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≠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)∈F[x] nekonstanten 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 ℤ_p, izomorfni. Iz teh trditev sledi karakterizacija končnih polj, ki pove, da za poljubno praštevilo p in poljuben n∈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:Karakteristika polja, karakteristika kolobarja, klasifikacija končnih polj, razširitev polja, faktorski kolobar, polje, končno polje, ideal, cel kolobar, maksimalni ideal in praideal, kolobar polinomov, homomorfizem kolobarjev, razpadno polje, vektorski prostor, Wedderburnov izrek, ničle polinoma.
Year of publishing:2018
Publisher:[A. Vok]
Source:Maribor
UDC:512.624(043.2)
COBISS_ID:23867912 Link is opened in a new window
License:CC BY-NC-ND 4.0
This work is available under this license: Creative Commons Attribution Non-Commercial No Derivatives 4.0 International
Views:141
Downloads:28
Metadata:XML RDF-CHPDL DC-XML DC-RDF
Categories:FNM
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Secondary language

Language:English
Title:Finite fields
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≠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)∈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 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.
Keywords:Characteristic of a field, characteristic of a ring, classification of finite field, extension field, factor ring, field, finite field, ideal, ideal domain, maximal ideal and prime ideal, polynomial ring, ring homomorphism, splitting field, vector space, Wedderburn's theorem, zeros of polynomial.


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