delo diplomskega seminarja
Abstract
Za kolobar $R$ definiramo komutirajoči graf $\Gamma(R)$ kot graf, v katerem so vozlišča necentralni elementi kolobarja $R$, dve vozlišči pa sta povezani natanko tedaj, ko pripadajoča elementa komutirata v $R$. Pokažemo, da je za kolobarje matrik nad poljem in $n \ge 3$, komutirajoči graf $\Gamma (M_n(F))$ povezan natanko tedaj, ko ima vsaka $F$-razširitev stopnje $n$ pravo vmesno polje. Nadalje pokažemo, da je $\Gamma (M_n(\mathbb{Q}))$ nepovezan $n \ge 2$. Dokažemo, da če je $\Gamma (M_n(F)))$ povezan, potem je njegov premer vsaj 4 in največ 6. Poiščemo nekaj primerov komutirajočih grafov s premerom 4. Dokažemo še, da če je $F$ končno polje in $n$ ni praštevilo ali kvadrat praštevila, je ${\rm diam}\,\Gamma (M_n(F)) \le 5$.
Keywords
matematika;komutirajoči graf;linearna algebra;matrike;Galoisova teorija;
Data
Language: |
Slovenian |
Year of publishing: |
2022 |
Typology: |
2.11 - Undergraduate Thesis |
Organization: |
UL FMF - Faculty of Mathematics and Physics |
Publisher: |
[L. Sajovic] |
UDC: |
512 |
COBISS: |
101306115
|
Views: |
851 |
Downloads: |
39 |
Average score: |
0 (0 votes) |
Metadata: |
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Other data
Secondary language: |
English |
Secondary title: |
The missing field |
Secondary abstract: |
We define the commuting graph of ring $R$ as the graph $\Gamma(R)$ in which vertices are non-central elements of ring $R$. Two vertices are adjacent if and only if the corresponding elements commute in $R$. We show that for the ring of matrices over a field where $n \ge 3$ the commuting graph $\Gamma (M_n(F))$ is connected if and only if for every $F$-extension of degree $n$ exists a proper intermediate field. We also show that $\Gamma (M_n(\mathbb{Q}))$ is not connected for $n \ge 2$. We prove that if $\Gamma (M_n(F))$ is connected then $4 \le {\rm diam}\,\Gamma (M_n(F)) \le 6$. We find some examples of commuting graphs with diameter 4. We also prove that ${\rm diam}\,\Gamma (M_n(F)) \le 5$ if $F$ is a finite field and $n$ is not a prime nor square of a prime. |
Secondary keywords: |
mathematics;commuting graph;linear algebra;matrices;Galois theory; |
Type (COBISS): |
Final seminar paper |
Study programme: |
0 |
Thesis comment: |
Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja |
Pages: |
29 str. |
ID: |
14785285 |