{"data":{"id":"10.13016/m22g7r","type":"dois","attributes":{"doi":"10.13016/m22g7r","prefix":"10.13016","suffix":"m22g7r","identifiers":[],"alternateIdentifiers":[],"creators":[{"name":"Hennessy, Angela Marie","nameType":"Personal","givenName":"Angela Marie","familyName":"Hennessy","affiliation":[],"nameIdentifiers":[]}],"titles":[{"title":"An Algorithmic Approach to Invariant Rational Functions"}],"publisher":"Digital Repository at the University of Maryland","container":{},"publicationYear":2014,"subjects":[{"subject":"Mathematics","subjectScheme":"pqcontrolled"},{"subject":"FOS: Mathematics","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subjectScheme":"pquncontrolled"}],"contributors":[],"dates":[{"date":"2014","dateType":"Issued"}],"language":"en","types":{"ris":"THES","bibtex":"phdthesis","citeproc":"thesis","schemaOrg":"Thesis","resourceType":"Dissertation","resourceTypeGeneral":"Collection"},"relatedIdentifiers":[],"relatedItems":[],"sizes":[],"formats":[],"version":null,"rightsList":[],"descriptions":[{"description":"For any field K and group A acting on K(x0, x1,..., xn-1), the fixed field consists of the elements fixed under the action of A, and is denoted K(x0, x1,..., xn-1)^A. Noether's problem seeks to answer whether K(x0, x1,..., xn-1)^A/K is a purely transcendental field extension.\n\nLenstra gave necessary and sufficient conditions for a purely transcendental extension when A is any finite abelian group, however his proof is often regarded as non-constructive. Masuda constructed generators of the fixed field when a certain ideal is principal in an integral group ring, and exhibited results when A is a cyclic group of order n \u0026lt;= 7 and n=11. We prove, however, that the ideal in question is not principal when n=13, 17, 19 or 23, and thus Masuda's techniques cannot be used. We use Lenstra's proof as a basis of an algorithm which computes generators of the fixed field. We demonstrate this algorithm for groups of order n=3, 5, 7, 11, 13, 17, 19 and 23.\n\nSwan gave the first negative answer to Noether's problem for the cyclic group of order 47 over Q. Leshin quantifies the degree to which a field extension fails to be purely transcendental by defining the degree of irrationality. We use the fact that the class group of the maximal real subfield Q(zeta_23+zeta_23^-1) is trivial to construct a field extension of degree 47 below the fixed field that is purely transcendental over K, thus providing a new bound for the degree of irrationality.","descriptionType":"Abstract"}],"geoLocations":[],"fundingReferences":[],"xml":"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","url":"http://hdl.handle.net/1903/16075","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-2.2","source":null,"isActive":true,"state":"findable","reason":null,"viewCount":0,"viewsOverTime":[],"downloadCount":0,"downloadsOverTime":[],"referenceCount":0,"citationCount":0,"citationsOverTime":[],"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2015-02-05T06:35:11.000Z","registered":"2015-02-05T06:35:12.000Z","published":"2014","updated":"2020-08-19T17:16:39.000Z"},"relationships":{"client":{"data":{"id":"umd.lib","type":"clients"}},"provider":{"data":{"id":"umd","type":"providers"}},"media":{"data":{"id":"10.13016/m22g7r","type":"media"}},"references":{"data":[]},"citations":{"data":[]},"parts":{"data":[]},"partOf":{"data":[]},"versions":{"data":[]},"versionOf":{"data":[]}}}}