Class numbers of cyclotomic function fields

Authors:
K. F. Ireland and R. D. Small

Journal:
Math. Comp. **46** (1986), 337-340

MSC:
Primary 11R58

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815854-5

MathSciNet review:
815854

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Abstract: 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.

**[1]**E. Artin, "Quadratische Körper im Gebiet der höheren Kongruenzen. I, II,"*Math. Z.*, v. 19, 1924, pp. 153-246. MR**1544651****[2]**S. Galovich & M. Rosen, "The class number of cyclotomic function fields,"*J. Number Theory*, v. 13, 1981, pp. 363-375. MR**634206 (83m:12022)****[3]**D. Hayes, "Explicit class field theory for rational function fields,"*Trans. Amer. Math. Soc.*, v. 189, 1979, pp. 77-91. MR**0330106 (48:8444)****[4]**K. Ireland & Michael Rosen,*A Classical Introduction to Number Theory*, Graduate Texts in Mathematics, Springer-Verlag, New York, Heidelberg, Berlin, 1982. MR**661047 (83g:12001)****[5]**K. Iwasawa, "A note on class numbers of algebraic number fields,"*Abh. Math. Sem. Univ. Hamburg*, v. 20, 1955, pp. 257-258. MR**0083013 (18:644d)****[6]**M. Rosen, "Ambiguous divisor classes in function fields,"*J. Number Theory*, v. 9, 1977, pp. 160-174. MR**0463142 (57:3102)**

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0815854-5

Article copyright:
© Copyright 1986
American Mathematical Society