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[{"key": "dc.contributor.advisor", "value": "Lehtonen, Ari", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Ojaniemi, Jenna", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-20T06:01:15Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-20T06:01:15Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/64737", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tutkielman tarkoituksena on Pellin yht\u00e4l\u00f6n ratkaiseminen ja aritmetiikan peruslauseen voimassaolon tutkiminen algebrallisten kokonaislukujen muodostamissa renkaissa \\mathbb{Z}[\\sqrt{-2}], \\mathbb{Z}[\\sqrt{-3}],\\mathbb{Z}[\\sqrt{\\zeta_3}]$ ja $\\mathbb{Z}[\\sqrt{-5}]. Aritmetiikan peruslauseella tarkoitetaan yleisimmin positiivisten kokonaislukujen yksik\u00e4sitteist\u00e4 alkulukuhajotelmaa. Pellin yht\u00e4l\u00f6n ratkaisussa k\u00e4ytetyt tavat k\u00e4sitell\u00e4 algebrallisia kokonaislukuja ovat apuna aritmetiikan peruslauseen yleist\u00e4misess\u00e4 muihin lukuluokkiin. Tutkielmassa tutustutaan my\u00f6s ketjumurtolukujen tarjoamaan ratkaisualgoritmiin Pellin yht\u00e4l\u00f6lle. Lis\u00e4ksi tutkielmassa k\u00e4sitell\u00e4\u00e4n ideaalien teoriaa, sill\u00e4 jos varsinaista m\u00e4\u00e4ritelm\u00e4n mukaista yksik\u00e4sitteist\u00e4 tekij\u00f6ihinjakoa ei pystyt\u00e4 renkaalle yleist\u00e4m\u00e4\u00e4n, voidaan alkutekij\u00e4hajotelmaa tarkastella alkuideaalien avulla.\n\nTutkielmassa aloitetaan algebran ja lukuteorian kurssilla k\u00e4sitellyist\u00e4 m\u00e4\u00e4ritelmist\u00e4 ja edet\u00e4\u00e4n asteittain vaativampiin algebrallisiin rakenteisiin. Tutkielmassa k\u00e4ytet\u00e4\u00e4n kuvia ja geometriaa algebrallisten todistusten rinnalla. Lis\u00e4ksi perehdyt\u00e4\u00e4n hieman k\u00e4sitelt\u00e4vien aiheiden historiaan sek\u00e4 tietokonelaskemiseen. Tutkielman kahdessa ensimm\u00e4isess\u00e4 luvussa k\u00e4yd\u00e4\u00e4n l\u00e4pi tutkielman kannalta t\u00e4rkeit\u00e4 tuloksia ja esitet\u00e4\u00e4n aritmetiikan peruslauseen todistus positiivisilla kokonaisluvuilla. Kolmas luku k\u00e4sittelee Pellin yht\u00e4l\u00f6\u00e4 ja nelj\u00e4s luku aritmetiikan peruslauseen yleist\u00e4mist\u00e4. Viidenness\u00e4 luvussa tutkitaan yksik\u00e4sitteisen tekij\u00f6ihinjaon ep\u00e4onnistumista ja perehdyt\u00e4\u00e4n ideaaleihin.\n\nTuloksena saadaan yksik\u00e4sitteisen tekij\u00f6ihinjaon onnistuminen renkaissa \\mathbb{Z}[\\sqrt{-2}] ja \\mathbb{Z}[\\zeta_3]. Yksik\u00e4sitteinen tekij\u00f6ihinjako ep\u00e4onnistuu renkaissa \\mathbb{Z}[\\sqrt{-3}] ja \\mathbb{Z}[\\sqrt{-5}]. Toisaalta renkaalle \\mathbb{Z}[\\sqrt{-5}] voidaan m\u00e4\u00e4ritt\u00e4\u00e4 alkutekij\u00e4hajotelma k\u00e4ytt\u00e4en alkuideaaleja, ja alkutekij\u00e4hajotelma on yksik\u00e4sitteinen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-06-20T06:01:15Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-20T06:01:15Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "65", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "aritmetiikan peruslause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "algebrallisten kokonaislukujen muodostama rengas", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Eukleideen alue", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ideaali", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "jakoyht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Pellin yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "p\u00e4\u00e4ideaalialue", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Algebrallista lukuteoriaa : Pellin yht\u00e4l\u00f6st\u00e4 ja aritmetiikan peruslauseen yleist\u00e4misest\u00e4", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201906203324", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": 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