Study of cyclotomic extensions of degree power of 2 and  classification of radical extensions up to isomorphism

Mrema, Elizabeth

Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism

dc.contributor.advisor	Marques, Sophie	en_ZA
dc.contributor.author	Mrema, Elizabeth	en_ZA
dc.contributor.other	Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.	en_ZA
dc.date.accessioned	2024-02-20T10:58:55Z
dc.date.accessioned	2024-04-26T13:12:37Z
dc.date.available	2024-02-20T10:58:55Z
dc.date.available	2024-04-26T13:12:37Z
dc.date.issued	2024-03
dc.description	Thesis (PhD)--Stellenbosch University, 2024.	en_ZA
dc.description.abstract	In this thesis, we gain a deeper understanding of cyclotomic extensions of degree powers of 2 and the classification of radical extensions (both separable and insepa‑ rable) up to isomorphism. Our main results about cyclotomic extensions of degree power of 2 describe their Galois structures, their degrees, their subextensions, their tower decompositions, and the minimal polynomials of some traces of root of unity generating all their subsextensions over an arbitrary base field. In exploring these as‑ pects, we discover two important invariants 𝓁𝑝∞ and 𝜈𝑝∞ where 𝑝 is a prime number, holding essential information about cyclotomic extensions of degree 2 and those gen‑ erated by primitive (2𝑒)𝑡ℎ roots of unity where 𝑒 ∈ ℕ. In our quest to provide explicit expressions for the coefficients of the minimal polynomials of the subextensions of cyclotomic extensions generated by primitive (2𝑒)𝑡ℎ root of unity, we discover fasci‑ nating characterizations, some of which are linked to the well‑known Catalan num‑ bers solving Combinatorial problems using field theory. Building upon the insights gained from our exploration of cyclotomic extensions, we provide a comprehensive classification of separable and inseparable radical ex‑ tensions up to isomorphism. In order to have a global understanding of these exten‑ sions up to isomorphism, we exhibit a meaningful parameterization of the set of iso‑ morphic radical extensions into moduli spaces involving the action of some groups.	en_ZA
dc.description.abstract	AFRIKAANSE OPSOMMING: In hierdie tesis verkry ons ’n dieper verstaan van siklotomiese uitbreidings waarvan die graad magte van 2 is en die klassifikasie van radikale uitbreidings (skeibaar en on‑ skeibaar), tot isomorfisme. Ons hoofresultate oor siklotomies uitbreidings waarvan die graad ŉ mag van 2 is, beskryf hulle Galois‑strukture, hulle grade, hulle deeluit‑ breidings, hulle toring‑ontbindings, en die minimale polinome van sommige spore van eenheidwortels wat al hul deeluitbreidings oor ’n willekeurige basisliggaam ge‑ nereer. Deur hierdie aspekte te ondersoek, ontdek ons twee belangrike invariante 𝓁𝑝∞ en 𝜈𝑝∞ waar 𝑝 ’n priemgetal is, wat noodsaaklike inligting bevat oor siklotomiese uitbreidings van graad 2 en dié gegenereer deur primitiewe (2𝑒)𝑑𝑒 eenheidswortels waar 𝑒 ∈ ℕ. In ons strewe om eksplisiete uitdrukkings te verskaf vir die koëffisi‑ ënte van die minimale polinome van die deeluitbreidings van siklotomiese uitbrei‑ dings gegenereer deur primitiewe (2𝑒)𝑑𝑒 eenheidswortels, ontdek ons fassinerende karakteriserings, waarvan sommige gekoppel is aan die bekende Katalaanse getalle wat kombinatoriese probleme oplos deur liggaamsteorie te gebruik. Voortbouend op die insigte verkry uit ons verkenning van siklotomiese uitbrei‑ dings, bied ons ’n omvattende klassifikasie van skeibare en onskeibare radikale uit‑ breidings, tot isomorfisme. Ten einde ’n globale verstaan van hierdie uitbreidings, tot isomorfisme, te hê, toon ons ’n betekenisvolle parameterisering van die stel iso‑ morfiese radikale uitbreidings in moduli‑ruimtes wat die aksie van sommige groepe behels.	af_ZA
dc.description.version	Doctorate	en_ZA
dc.identifier.uri	https://scholar.sun.ac.za/handle/10019.1/130317
dc.language.iso	en_ZA	en_ZA
dc.language.iso	en_ZA	en_ZA
dc.publisher	Stellenbosch : Stellenbosch University	en_ZA
dc.rights.holder	Stellenbosch University	en_ZA
dc.subject.lcsh	Field extensions (Mathematics)	en_ZA
dc.subject.lcsh	Cyclotomic fields	en_ZA
dc.subject.lcsh	Isomorphisms (Mathematics)	en_ZA
dc.subject.lcsh	Galois modules (Algebra)	en_ZA
dc.subject.name	UCTD	en_ZA
dc.title	Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism	en_ZA
dc.type	Thesis	en_ZA

Files

Original bundle

Now showing 1 - 1 of 1

Name:: mrema_study_2024.pdf
Size:: 2.25 MB
Format:: Adobe Portable Document Format
Description:

Download

Collections

Doctoral Degrees (Mathematical Sciences)