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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": null, "element": "contributor", "schema": "dc", "qualifier": "advisor"}, {"key": "dc.contributor.author", "value": "Kuoppam\u00e4ki, Miia", "language": null, "element": "contributor", "schema": "dc", "qualifier": "author"}, {"key": "dc.date.accessioned", "value": "2025-05-23T05:36:43Z", "language": null, "element": "date", "schema": "dc", "qualifier": "accessioned"}, {"key": "dc.date.available", "value": "2025-05-23T05:36:43Z", "language": null, "element": "date", "schema": "dc", "qualifier": "available"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "schema": "dc", "qualifier": "issued"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/102706", "language": null, "element": "identifier", "schema": "dc", "qualifier": "uri"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tutustutaan polynomiyht\u00e4l\u00f6iden ratkaisemiseen ja n\u00e4iden ratkaisuiden historiaan. Tarkoituksena on esitell\u00e4 historiaa muinaisesta Egyptist\u00e4 ja Babyloniasta aina 1800-luvun alkuun asti. Tutkielman tarkoituksena on luoda selke\u00e4 kokonaiskuva ratkaisuista. Tutkielman alussa esitell\u00e4\u00e4n joitakin tutkielmassa k\u00e4ytett\u00e4vi\u00e4 esitietoja, jonka j\u00e4lkeen siirryt\u00e4\u00e4n ensimm\u00e4isen asteen polynomiyht\u00e4l\u00f6ihin. Tutkielmassa esitell\u00e4\u00e4nkin muun muassa esimerkki muinaisen Egyptin ratkaisutavasta ensimm\u00e4isen asteen yht\u00e4l\u00f6ille. T\u00e4m\u00e4n j\u00e4lkeen siirryt\u00e4\u00e4n toisen asteen polynomiyht\u00e4l\u00f6ihin, tutustumalla sek\u00e4 historiaan ett\u00e4 kahteen ratkaisutapaan. Ratkaisutavoista ensimm\u00e4inen seuraa Al-Khw\u00e2rizmin geometriaa hy\u00f6dynt\u00e4v\u00e4\u00e4 ratkaisua, kun taas j\u00e4lkimm\u00e4inen esittelee algebrallisen ratkaisun yleiselle toisen asteen polynomiyht\u00e4l\u00f6lle. Seuraavaksi vuorossa ovat kolmannen asteen polynomiyht\u00e4l\u00f6t. Historian lis\u00e4ksi esitell\u00e4\u00e4n my\u00f6s kaksi versiota ratkaisulle. Ratkaisuista ensimm\u00e4inen seuraa jonkin verran Cardanon alkuper\u00e4ist\u00e4 ratkaisua, kun taas j\u00e4lkimm\u00e4inen hy\u00f6dynt\u00e4\u00e4 kompleksilukuja. Kolmannen asteen polynomeihin tutustumisen j\u00e4lkeen siirryt\u00e4\u00e4n nelj\u00e4nnen asteen polynomeihin. My\u00f6s n\u00e4iden kohdalla esitell\u00e4\u00e4n kaksi ratkaisutapaa, joista ensimm\u00e4inen on modernimpi ja j\u00e4lkimm\u00e4inen seuraa jonkin verran Ferrarin alkuper\u00e4ist\u00e4 ratkaisua. T\u00e4m\u00e4n j\u00e4lkeen kerrotaan lyhyesti viidennen ja korkeamman asteen polynomiyht\u00e4l\u00f6iden ratkeamattomuudesta, sek\u00e4 pohditaan miksi kolmannen ja nelj\u00e4nnen asteen yht\u00e4l\u00f6iden ratkaisuissa k\u00e4ytetty resolventtiyht\u00e4l\u00f6 ei n\u00e4iden ratkaisuissa toimi. Tutkielman loppu k\u00e4sittelee juurien reaalisuustarkasteluja, jossa tutustutaan kolmannen ja nelj\u00e4nnen asteen yht\u00e4l\u00f6iden reaalisten juurien lukum\u00e4\u00e4riin.", "language": "fi", "element": "description", "schema": "dc", "qualifier": "abstract"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "schema": "dc", "qualifier": "abstract"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-05-23T05:36:43Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "schema": "dc", "qualifier": "provenance"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-05-23T05:36:43Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "schema": "dc", "qualifier": "provenance"}, {"key": "dc.format.extent", "value": "41", "language": null, "element": "format", "schema": "dc", "qualifier": "extent"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "schema": "dc", "qualifier": "iso"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "schema": "dc", "qualifier": null}, {"key": "dc.subject.other", "value": "", "language": null, "element": "subject", "schema": "dc", "qualifier": "other"}, {"key": "dc.title", "value": "Polynomiyht\u00e4l\u00f6iden ratkaisemisen historiaa", "language": null, "element": "title", "schema": "dc", "qualifier": null}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "schema": "dc", "qualifier": null}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202505234541", "language": null, "element": "identifier", "schema": "dc", "qualifier": "urn"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "schema": "dc", "qualifier": "faculty"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "schema": "dc", "qualifier": "faculty"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "schema": "dc", "qualifier": "department"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "schema": "dc", "qualifier": "department"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": null, "element": "contributor", "schema": "dc", "qualifier": "organization"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "element": "contributor", "schema": "dc", "qualifier": "organization"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "schema": "dc", "qualifier": "discipline"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "schema": "dc", "qualifier": "discipline"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "schema": "dc", "qualifier": "coar"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": "fi", "element": "rights", "schema": "dc", "qualifier": "copyright"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "schema": "dc", "qualifier": "accesslevel"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "schema": "dc", "qualifier": "publication"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "schema": "dc", "qualifier": "yso"}, {"key": "dc.subject.yso", "value": "polynomit", "language": null, "element": "subject", "schema": "dc", "qualifier": "yso"}, {"key": "dc.subject.yso", "value": "yht\u00e4l\u00f6t", "language": null, "element": "subject", "schema": "dc", "qualifier": "yso"}, {"key": "dc.subject.yso", "value": "algebra", "language": null, "element": "subject", "schema": "dc", "qualifier": "yso"}, {"key": "dc.subject.yso", "value": "historia", "language": null, "element": "subject", "schema": "dc", "qualifier": "yso"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "schema": "dc", "qualifier": "url"}]
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