Class numbers of cyclotomic function fields
Class numbers of cyclotomic function fields
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DOI：
10.2307/2008237
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发表时间：
1986
影响因子：
2
通讯作者：
K. Ireland;R. D. Small
中科院分区：
文献类型：
--
作者：
K. Ireland;R. D. Small
Using results of Galovich and Rosen the plus and minus factors for the class number of the cyclotomic function field associated with irreducibles of degree three and four over the-field with three elements are computed. As a consequence it is shown that the analogue to a result of Kummer on the p-divisibility of these factors is false. Consider an odd prime p and let F denote the finite field with p elements. To each monic irreducible polynomial P(T) of degree d with coefficients in F one can associate a cyclic extension Kp of F (T) that enjoys properties analogous to those of the classical cyclotomic number fields Q(tP), p = e2' /7. Such fields were dis- covered by L. Carlitz in the 1930's and have been studied intensively in recent years by D. Hayes, M. Rosen, S. Galovich, D. Goss and others. If D is the integral closure of F (T) in Kp, then one has the diagram:
