Let K/Fq be an algebraic function field with full constant field Fq and genus g. Then
the divisor class number hK of K/Fq is the order of the quotient group, D0K
/P(K),
degree zero divisors of K over principal divisors of K. The classification of the function
fields K with hK = 1 is done by MacRea, Leitzel, Madan and Queen and the classification
of the extensions with class number two is done by Le Brigand. Determination
of the necessary and the sufficient conditions for a function field to have class number
three is done by H¨ / ulya T¨ / ore.
Let k := Fq(T) be the rational function field over the finite field Fq with q elements.
For a polynomial N &isin / Fq[T], we construct the Nth cyclotomic function field KN.
Cyclotomic function fields were investigated by Carlitz, studied by Hayes, M. Rosen,
M. Bilhan and many other mathematicians. Classification of cyclotomic function
fields and subfields of cyclotomic function fields with class number one is done by
Kida, Murabayashi, Ahn and Jung. Also the classification of function fields with
genus one and classification of those with class number two is done by Ahn and Jung.
In this thesis, we classified all algebraic function fields and subfields of cyclotomic function fields over finite fields with class number three.
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/12613080/index.pdf |
| Date | 01 February 2011 |
| Creators | Buyruk, Dilek |
| Contributors | Bilhan, Mehpare |
| Publisher | METU |
| Source Sets | Middle East Technical Univ. |
| Language | English |
| Detected Language | English |
| Type | Ph.D. Thesis |
| Format | text/pdf |
| Rights | To liberate the content for METU campus |
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