Manjkajoči obseg : delo diplomskega seminarja

Sajovic, Luka (author), Dolžan, David (mentor)

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 ${\rm diam}\,\Gamma (M_n(F)) \le 5$.

• Language: Slovenian
• Keywords: komutirajoči graf, linearna algebra, matrika, Galoisova teorija
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2022
• PID: 20.500.12556/RUL-135507
• UDC: 512
• COBISS.SI-ID: 101306115
• Publication date in RUL: 17.03.2022

Secondary language

• Language: English
• Title: The missing field

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.

• Keywords: commuting graph, linear algebra, matrix, Galois theory