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Solving embedding problems with bounded ramification
Thesis (PhD)--Stellenbosch University, 2016
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2016
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| _version_ | 1867613895721484288 |
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| access_status_str | Open Access |
| author | Ramiharimanana, Nantsoina Cynthia |
| author2 | Green, Barry |
| author_browse | Green, Barry Ramiharimanana, Nantsoina Cynthia |
| author_facet | Green, Barry Ramiharimanana, Nantsoina Cynthia |
| author_sort | Ramiharimanana, Nantsoina Cynthia |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2016 |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/100049 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:43:25.190Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/100049 Solving embedding problems with bounded ramification Ramiharimanana, Nantsoina Cynthia Green, Barry Jarden, Moshe Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences Galois group Epimorphism Galois extension Homomorphism UCTD Thesis (PhD)--Stellenbosch University, 2016 ENGLISH ABSTRACT : Given a Galois extension K/K0 of number fields, a finite group G, and an epimorphism α: G→ Gal(K/K0) with solvable kernel, our goal is to embed K into a Galois extension N of K0 with Galois group Gal(N/K0) ≅ G such that the restriction map resN=K: Gal(N/K0) → Gal(K/K0) coincides with α and │Ram(N/K0)│ ≤│ Ram(K│K0) │+ Ω (│Ker(α)│). Here Ram(N│K0) is the finite set of primes of K0 that ramify in N and Ω (│Ker (α) │) is the number of the prime divisors, counted with multiplicity, of │Ker (α). We achieve our goal under two conditions: first, the number of roots of unity in K should be relatively prime to the order of Ker(α). The second one demands that each local embedding problem resulted from the original one should be "weakly solvable". In fact, our solution locally coincides with finitely many "local weak solutions" given in advance. Our result strengthens a former result of Neukirch in [Neu79], where the same embedding problem satisfying the same conditions is solved without giving a bound on the ramification. In particular, the above mentioned local conditions are satisfied if the epimorphism α has a section. This leads to a well known result of Shafarevich that does not assume the condition on the roots of unity but pays with a huge number of ramified primes (that appears when one analyses Shafarevich's proof). Like in [Neu79], our proof uses class field theory in its cohomological approach. The bounding of the ramification is based, in addition to the above mentioned tools, on a strengthening of a lemma of [GeJ98]. AFRIKAANSE OPSOMMING : Laat K/K0 'n Galois uitbreiding van getalleliggame wees, G 'n eindige groep, en α: G→ Gal(K/K0) 'n epimorfisme met oplosbare kern. Ons doel is om K in 'n Galois uitbreiding N van K0 in-te-bed sodat die Galois groep Gal(N/K0) ≅ G; en sodat die beperkingsafbeelding resN=K: Gal(N/K0) → Gal(K/K0) ooreenstem met α en │Ram(N=K0) │ ≤│+Ram(K/K0)+ Ω (│Ker(α ) │). Hier is Ram(N/K0) die eindige versameling van priemdelers van K0 wat in N vertak, en Ω (│Ker(α) │) is die aantal priemdelers van │Ker(α) │, getel met multiplisiteit. Ons bereik hierdie doelstelling onderhewig aan twee voorwaardes: Eerstens moet die aantal wortels van eenheid in K relatief priem wees aan die orde van Ker (α). Tweedens eis ons dat elke lokale inbeddingsprobleem, wat volg uit die oorspronklike een, "swak oplosbaar" moet wees. Meer presies gestel, sal ons oplossing lokaal ooreenstem met 'n eindige aantal "lokaal swak oplossings" wat vooraf gegee word. Ons resultaat versterk 'n vroeer resultaat van Neukirch in [Neu79], waar 'n inbeddingsprobleem wat dieselfde voorwaardes bevredig opgelos word, maar sonder die grens op die aantal vertakkings. In die besonder word die lokale voorwaardes bevredig mits die epimorfisme α 'n snitafbeelding besit. Hieruit volg dan ook die bekende resultaat van Shafarevich, wat nie die voorwaarde oor die wortels van eenheid benodig nie, maar gevolglik 'n baie groot aantal priemdelers wat vertak veroorsaak (hierdie opmerking word gesien wanneer sy bewys in detail bestudeer word). Soos in [Neu79], maak ons gebruik van klasliggaamteorie met 'n kohomologiese benadering. Die begrensdheid van die aantal priemdelers wat vertak maak ook gebruik van 'n versterking van 'n hulpstelling uit [GeJ98]. Doctoral 2016-12-22T13:09:25Z 2016-12-22T13:09:25Z 2016-12 Thesis http://hdl.handle.net/10019.1/100049 en_ZA Stellenbosch University viii, 84 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Galois group Epimorphism Galois extension Homomorphism UCTD Ramiharimanana, Nantsoina Cynthia Solving embedding problems with bounded ramification |
| title | Solving embedding problems with bounded ramification |
| title_full | Solving embedding problems with bounded ramification |
| title_fullStr | Solving embedding problems with bounded ramification |
| title_full_unstemmed | Solving embedding problems with bounded ramification |
| title_short | Solving embedding problems with bounded ramification |
| title_sort | solving embedding problems with bounded ramification |
| topic | Galois group Epimorphism Galois extension Homomorphism UCTD |
| url | http://hdl.handle.net/10019.1/100049 |
| work_keys_str_mv | AT ramiharimanananantsoinacynthia solvingembeddingproblemswithboundedramification |