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[{"key": "dc.contributor.advisor", "value": "Eriksson-Bique, Sylvester", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Viitala, Mia", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-06-06T07:11:54Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-06-06T07:11:54Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/95579", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Ta\u0308ma\u0308n pro gradu -tutkielman tarkoituksena on perehtya\u0308 transkendenttilukuihin seka\u0308 algebrallisiin lukuihin. Algebrallinen luku on jonkin rationaalilukukertoimisen polynomin juuri. Jos luku ei ole algebrallinen niin se on transkendentti. Tutkielmassa aihetta la\u0308hestyta\u0308a\u0308n kuntien, kuntalaajennosten seka\u0308 symmetristen polynomien kautta. Tutkielman tarkoituksena on antaa tiivis yleiskuvaus aiheesta.\n\nEnsin tutkielmassa perehdyta\u0308a\u0308n pohjatietoina rationaali- ja irrationaalilukuihin seka\u0308 erityisesti esiteta\u0308a\u0308n todistukset lukujen \u03c0 ja e irrationaalisuudesta luvussa 1. Pohjatietojen ka\u0308sittelyn ja\u0308lkeen annetaan ma\u0308a\u0308ritelma\u0308 transkendenttiluvulle ja algebralliselle luvulle. Ta\u0308sta\u0308 siirryta\u0308a\u0308n tarkastelemaan tutkielmassa myo\u0308hemmin tarvittavaa perusalgebraa luvussa 2. Luvussa 3 perehdyta\u0308a\u0308n polynomirenkaisiin seka\u0308 symmetrisiin polynomeihin. Samassa luvussa todistetaan symmetristen polynomien peruslause, jolla on oleellinen osa tutkielman pa\u0308a\u0308tuloksien todistuksissa.\n\nLuvussa 4 tutustutaan kuntalaajennoksiin ja erilaisiin niita\u0308 koskeviin tuloksiin. Luvussa 5 ka\u0308sitella\u0308a\u0308n algebrallisten lukujen ominaisuuksia ja osoitetaan, etta\u0308 algebralliset luvut muodostavat kunnan. Tutkielman pa\u0308a\u0308tuloksina luvussa 6 esiteta\u0308a\u0308n todistukset lukujen \u03c0 ja e transkendenttiudesta. Tutkielman lopuksi luvussa 6 esitella\u0308a\u0308n viela\u0308, miten luvun \u03c0 transkendenttius liittyy antiikin konstruktio-ongelmaan ympyra\u0308n nelio\u0308innista\u0308 ja samassa luvussa esitella\u0308a\u0308n myo\u0308s kuution kahdentamiseen liittyva\u0308 konstruktio-ongelma. Ta\u0308ma\u0308n lisa\u0308ksi otetaan pieni katsaus siihen millaisiin transkendenttilukuihin liittyviin ratkaisemattomiin ongelmiin nykypa\u0308iva\u0308n matemaatikot yritta\u0308va\u0308t lo\u0308yta\u0308a\u0308 ratkaisuja.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-06-06T07:11:54Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-06-06T07:11:54Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "45", "language": "", "element": "format", "qualifier": "extent", "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": "transkendenttiluku", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "algebrallinen luku", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kuntalaajennos", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "konstruktio-ongelma", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "symmetriset polynomit", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kunnat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Transkendenttiluvut", "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-202406064339", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "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": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "polynomit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "irrationaaliluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "renkaat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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