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Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism
Thesis (PhD)--Stellenbosch University, 2024.
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| Format: | Thesis |
| Language: | en_ZA en_ZA |
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Stellenbosch : Stellenbosch University
2024
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| _version_ | 1867613956939448320 |
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| access_status_str | Open Access |
| author | Mrema, Elizabeth |
| author2 | Marques, Sophie |
| author_browse | Marques, Sophie Mrema, Elizabeth |
| author_facet | Marques, Sophie Mrema, Elizabeth |
| author_sort | Mrema, Elizabeth |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2024. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/130317 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA en_ZA |
| last_indexed | 2026-06-10T12:44:23.606Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/130317 Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism Mrema, Elizabeth Marques, Sophie Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Field extensions (Mathematics) Cyclotomic fields Isomorphisms (Mathematics) Galois modules (Algebra) UCTD Thesis (PhD)--Stellenbosch University, 2024. 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. 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. Doctorate 2024-02-20T10:58:55Z 2024-04-26T13:12:37Z 2024-02-20T10:58:55Z 2024-04-26T13:12:37Z 2024-03 Thesis https://scholar.sun.ac.za/handle/10019.1/130317 en_ZA en_ZA Stellenbosch University application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Field extensions (Mathematics) Cyclotomic fields Isomorphisms (Mathematics) Galois modules (Algebra) UCTD Mrema, Elizabeth Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism |
| title | Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism |
| title_full | Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism |
| title_fullStr | Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism |
| title_full_unstemmed | Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism |
| title_short | Study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism |
| title_sort | study of cyclotomic extensions of degree power of 2 and classification of radical extensions up to isomorphism |
| topic | Field extensions (Mathematics) Cyclotomic fields Isomorphisms (Mathematics) Galois modules (Algebra) UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/130317 |
| work_keys_str_mv | AT mremaelizabeth studyofcyclotomicextensionsofdegreepowerof2andclassificationofradicalextensionsuptoisomorphism |