Data · collection · 2014
An Algorithmic Approach to Invariant Rational Functions
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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.
Description
Lenstra 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 <= 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.
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We demonstrate this algorithm for groups of order n=3, 5, 7, 11, 13, 17, 19 and 23. Swan 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.
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Where it is published
- Repository landing page hdl.handle.net/1903/16075 ↗
landing page · from DataCite
- DOI doi.org/10.13016/m22g7r ↗
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- DataCite API api.datacite.org/dois/10.13016/m22g7r ↗
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Topics
- Stated by source
- Mathematics
- From keywords
- Mathematics & Statistics
- Inferred from text
- Group theory and generalisations 79%
Provenance · 1 source records, 6 field assertions
| Source | Key | Last seen | Raw |
|---|---|---|---|
| DataCite | 10.13016/m22g7r | 14 d ago | JSON v1 |
| Field | Assertion | Extractor | Evidence |
|---|---|---|---|
| concepts[field].anzsrc:field:490405 | enrichment · DataCite | taxonomy-embedding@1.1.0 | title+keywords+description (79%) |
| concepts[field].fos:mathematics | source · DataCite | connector:datacite@1.0.0 | |
| concepts[field].local:field:mathematics-statistics | mapping · DataCite | vocabulary-mapper@1.0.0 | keywords['Mathematics'] |
| description | source · DataCite | connector:datacite@1.0.0 | /data/attributes/descriptions |
| publication_date | source · DataCite | connector:datacite@1.0.0 | /data/attributes/dates |
| title | source · DataCite | connector:datacite@1.0.0 | /data/attributes/titles/0/title |