On algebraic function fields with class number three

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2011
Buyruk, Dilek
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 Ḧulya T̈ore. Let k := Fq(T) be the rational function field over the finite field Fq with q elements. For a polynomial N ∈ 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.