Calculation of the class numbers of imaginary cyclic quartic fields

Authors:
Kenneth Hardy, R. H. Hudson, D. Richman, Kenneth S. Williams and N. M. Holtz

Journal:
Math. Comp. **49** (1987), 615-620

MSC:
Primary 11Y40; Secondary 11R16, 11R29

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906194-5

MathSciNet review:
906194

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Abstract: Any imaginary cyclic quartic field can be expressed uniquely in the form $K = Q(\sqrt {A(D + B\sqrt D )} )$, where *A* is squarefree, odd and negative, $D = {B^2} + {C^2}$ is squarefree, $B > 0,C > 0$, and $(A,D) = 1$. Explicit formulae for the discriminant and conductor of *K* are given in terms of *A, B, C, D*. The calculation of tables of the class numbers $h(K)$ of such fields *K* is described.

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*Elementary number theory. An algebraic approach*, W. A. Benjamin, Inc., New York, 1970. MR**0252310** - Harvey Cohn,
*A computation of some bi-quadratic class numbers*, Math. Tables Aids Comput.**12**(1958), 213–217. MR**100972**, DOI https://doi.org/10.1090/S0025-5718-1958-0100972-2
M.-N. Gras, - Kenneth Hardy, R. H. Hudson, D. Richman, Kenneth S. Williams, and N. M. Holtz,
*Calculation of the class numbers of imaginary cyclic quartic fields*, Math. Comp.**49**(1987), no. 180, 615–620. MR**906194**, DOI https://doi.org/10.1090/S0025-5718-1987-0906194-5
R. H. Hudson & K. S. Williams,

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*A Class Number Formula for Certain Quartic Fields*, Carleton Mathematical Series No. 174, February 1981, 25 pp. B. Oriat,

*Groupe des Classes des Corps Quadratiques Réels*$Q(\sqrt d ),d < 10000$, Faculté des Sciences de Besançon, Besançon, France, 53 pp.

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Additional Information

Keywords:
Imaginary cyclic quartic fields,
class number,
discriminant

Article copyright:
© Copyright 1987
American Mathematical Society