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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Lindqvist, Ellinoora", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-26T06:37:17Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-26T06:37:17Z", "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/123456789/64858", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on n\u00e4ytt\u00e4\u00e4, kuinka kokonaislukujen lukuteoriaa voidaan yleist\u00e4\u00e4 kokonaislukujen kompleksisille laajennuksille. Lis\u00e4ksi halutaan osoittaa, ett\u00e4 tilannetta voidaan tarkastella toisestakin suunnasta eli siit\u00e4, kuinka kokonaislukujen lukuteorian tuloksia voidaan todistaa kompleksilukujen avulla. Kokonaislukujen kompleksisista laajennuksista erityisen kiinnostuneita tutkielmassa ollaan Gaussin, Hurwitzin ja Eisensteinin kokonaisluvuista.\n\nTutkielman alussa esitell\u00e4\u00e4n perusteellisesti Gaussin kokonaisluvut, jotka ovat t\u00e4rkein tutkielmassa k\u00e4ytett\u00e4vist\u00e4 kokonaislukujen laajennuksista. Gaussin kokonaisluvuille n\u00e4ytet\u00e4\u00e4n ensin niiden algebrallisia ominaisuuksia, mink\u00e4 j\u00e4lkeen siirryt\u00e4\u00e4n lukuteoreettisiin ominaisuuksiin. Osoittautuu, ett\u00e4 monet kokonaislukujen lukuteorian k\u00e4sitteet ja tulokset, kuten alkuluvut, Eukleideen algoritmi, aritmetiikan peruslause ja B\u00e9zout'n lemma, voidaan yleist\u00e4\u00e4 Gaussin kokonaisluvuille. \n\nT\u00e4m\u00e4n j\u00e4lkeen vaihdetaan tarkastelusuuntaa. Tutustutaan kokonaislukujen lukuteorian tuloksiin ja niiden todistamiseen kokonaislukujen kompleksilaajennusten avulla. Ensimm\u00e4isen\u00e4 annetaan tulos luonnollisen luvun esitt\u00e4misest\u00e4 kahden neli\u00f6n summana. Tuloksen todistuksessa hy\u00f6dynnet\u00e4\u00e4n Gaussin kokonaislukuja. T\u00e4m\u00e4n j\u00e4lkeen esitell\u00e4\u00e4n toinen kokonaislukujen kompleksisista laajennuksista, Hurwitzin kokonaisluvut. Niiden avulla todistetaan tulos luonnollisen luvun esitt\u00e4misest\u00e4 nelj\u00e4n neli\u00f6n summana. Kolmantena todistettavana lukuteorian tuloksena esitell\u00e4\u00e4n er\u00e4s tapaus Fermat'n suuresta lauseesta. Todistusta varten perehdyt\u00e4\u00e4n Eisensteinin kokonaislukuihin, jotka ovat viimeinen tutkielmassa esitelt\u00e4vist\u00e4 kokonaislukujen laajennuksista.\n\nLopuksi kootaan yhteen tutkielmassa k\u00e4ytetyt kokonaislukujen laajennukset niiden geometrisen tulkinnan kautta. Samalla k\u00e4sitell\u00e4\u00e4n lyhyesti tavallisten kokonaislukujen ja niiden kompleksisten laajennusten ulottuvuuksien m\u00e4\u00e4r\u00e4\u00e4 ja esitell\u00e4\u00e4n er\u00e4s siihen liittyv\u00e4 toistaiseksi ratkaisematon ongelma.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-06-26T06:37:17Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-26T06:37:17Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "48", "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": "Gaussin kokonaisluvut", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Hurwitzin kokonaisluvut", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Eisensteinin kokonaisluvut", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Gaussin alkuluvut", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "neli\u00f6iden summat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Fermat'n suuri lause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Kompleksilukujen lukuteoriaa ja lukuteoriaa kompleksiluvuilla", "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-201906263454", "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": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lukuteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "kokonaisluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "alkuluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algoritmit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "luonnolliset luvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "luvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "kompleksiluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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