Class numbers of pure fields

Author:
R. A. Mollin

Journal:
Proc. Amer. Math. Soc. **98** (1986), 411-414

MSC:
Primary 11R29; Secondary 11R21, 11R32

DOI:
https://doi.org/10.1090/S0002-9939-1986-0857932-2

MathSciNet review:
857932

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Necessary and sufficient conditions are given for the class number of a pure field to be divisible by for a given positive integer and prime . Moreover the divisibility of by is linked with the -rank of the class group of the and prime divisors of , where is a primitive th root of unity.

Finally we prove in an easy fashion that for a given odd prime and any natural number there exist infinitely many non-Galois algebraic number fields (in fact pure fields) of degree over whose class numbers are all divisible by .

**[1]**G. Cornell,*On the construction of the relative genus field*, Trans. Amer. Math. Soc.**271**(1982), 501-511. MR**654847 (83h:12019)****[2]**G. Cornell and M. Rosen,*Group theoretic constraints in the structure of the class group*, J. Number Theory**13**(1981), 1-11. MR**602445 (82e:12005)****[3]**T. Honda,*Pure cubic fields whose class numbers are multiples of three*, J. Number Theory**3**(1971), 7-12. MR**0292795 (45:1877)****[4]**K. Iimura,*A criterion for the class number of a pure quintic field to be divisible by*5, J. Reine Angew. Math.**292**(1976), 201-210. MR**0439806 (55:12688)****[5]**M. Ishida,*A note on class numbers of algebraic number fields*, J. Number Theory**1**(1969), 65-69. MR**0242788 (39:4115)****[6]**K. Iwasawa,*A note on class numbers of algebraic number fields*, Abh. Math. Sem. Univ. Hamburg**20**(1956), 257-258. MR**0083013 (18:644d)****[7]**R. Mollin,*Class numbers and a generalized Fermat theorem*, J. Number Theory**16**(1983), 420-429. MR**707613 (85f:11077a)****[8]**C. J. Parry*Class number relations in pure quintic fields*, Symposia Math.**15**(1975), 475-485. MR**0387241 (52:8084)****[9]**-,*Class number relations in pure sextic fields*, J. Reine Angew. Math.**274**/**275**(1975), 360-375. MR**0376614 (51:12789)****[10]**-,*Pure quartic number fields whose class numbers are even*, J. Reine Angew. Math.**264**(1975), 102-112. MR**0364181 (51:436)****[11]**C. J. Parry and C. D. Walter,*The class number of pure fields of prime degree*, Mathematika**23**(1976), 220-226. MR**0435032 (55:7994)****[12]**C. D. Walter,*Pure fields of degree*9*with class number prime to*3, Ann. Inst. Fourier Grenoble**30**(1980), 1-15. MR**584268 (82b:12006)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
11R29,
11R21,
11R32

Retrieve articles in all journals with MSC: 11R29, 11R21, 11R32

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0857932-2

Article copyright:
© Copyright 1986
American Mathematical Society