Pure fields of degree 9 with class number prime to 3

Colin D. Walter

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 2, page 1-15
  • ISSN: 0373-0956

Abstract

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The main theorem gives necessary conditions and sufficient conditions for Q ( n 9 ) to have class number prime to 3. These conditions involve only the rational prime factorization of n and congruences mod 27 of the prime factors of n . They give necessary and sufficient conditions for most n .

How to cite

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Walter, Colin D.. "Pure fields of degree 9 with class number prime to 3." Annales de l'institut Fourier 30.2 (1980): 1-15. <http://eudml.org/doc/74448>.

@article{Walter1980,
abstract = {The main theorem gives necessary conditions and sufficient conditions for $\{\bf Q\}(\@root 9 \of \{n\})$ to have class number prime to 3. These conditions involve only the rational prime factorization of $n$ and congruences mod 27 of the prime factors of $n$. They give necessary and sufficient conditions for most $n$.},
author = {Walter, Colin D.},
journal = {Annales de l'institut Fourier},
keywords = {Fields of Degree 9; Class Number},
language = {eng},
number = {2},
pages = {1-15},
publisher = {Association des Annales de l'Institut Fourier},
title = {Pure fields of degree 9 with class number prime to 3},
url = {http://eudml.org/doc/74448},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Walter, Colin D.
TI - Pure fields of degree 9 with class number prime to 3
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 2
SP - 1
EP - 15
AB - The main theorem gives necessary conditions and sufficient conditions for ${\bf Q}(\@root 9 \of {n})$ to have class number prime to 3. These conditions involve only the rational prime factorization of $n$ and congruences mod 27 of the prime factors of $n$. They give necessary and sufficient conditions for most $n$.
LA - eng
KW - Fields of Degree 9; Class Number
UR - http://eudml.org/doc/74448
ER -

References

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  1. [1] P. BARRUCAND and H. COHN, Remarks on principal factors in a relative cubic field, J. Number Theory, 3 (1971), 226-239. Zbl0218.12002MR43 #1945
  2. [2] R. BRAUER, Beziehungen zwischen Klassenzahlen von Teilkörpern eines galoisschen Körpers, Math. Nachr., 4 (1951), 158-174. Zbl0042.03801MR12,593b
  3. [3] A. ENDÔ, On the divisibility of the class number of Q(√9n) by 3, Mem. Fac. Sci., Kyushu Univ., A, 30 (1976), 299-311. Zbl0352.12008MR54 #5181
  4. [4] H. HASSE, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, II, Physica-Verlag, Würzburg/Wien, 1970. 
  5. [5] T. HONDA, Pure cubic fields whose class numbers are multiples of three, J. Number Theory, 3 (1971), 7-12. Zbl0222.12004MR45 #1877
  6. [6] K. IWASAWA, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 257-258. Zbl0074.03002MR18,644d
  7. [7] C. J. PARRY, Class number relations in pure quintic fields, Symposia Mathematica, 15 (1975), 475-485. Zbl0331.12003MR52 #8084
  8. [8] C. D. WALTER, A class number relation in Frobenius extensions of number fields, Mathematika, 24 (1977), 216-225. Zbl0358.12004MR57 #12458
  9. [9] C. D. WALTER, Kuroda's class number relation, Acta Arithmetica, 35 (1979), 41-51. Zbl0339.12008MR82d:12007
  10. [10] C. D. WALTER, The ambiguous class group and the genus group of certain non-normal extensions, Mathematika, 26 (1979), 113-124. Zbl0408.12004MR81c:12008

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