Computation of the ideal class group of certain complex quartic fields. II

Author:
Richard B. Lakein

Journal:
Math. Comp. **29** (1975), 137-144

MSC:
Primary 12A30; Secondary 12A50

MathSciNet review:
0444605

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Abstract: For quartic fields , where and is a prime of , the ideal class group is calculated by the same method used previously for quadratic extensions of , but using Hurwitz' complex continued fraction over . The class number was found for 10000 such fields, and the previous computation over was extended to 10000 cases. The distribution of class numbers is the same for 10000 fields of each type: real quadratic, quadratic over , quadratic over . Many fields were found with non-cyclic class group, including the first known real quadratics with groups and . Further properties of the continued fractions are also discussed.

**[1]**A. Hurwitz,*Über die Entwicklung complexer Grössen in Kettenbrüche*, Acta Math.**11**(1887), no. 1-4, 187–200 (German). MR**1554754**, 10.1007/BF02418048**[2]**S. KURODA, "Table of class numbers for quadratic fields ,"*Math. Comp.*, v. 29, 1975, UMT, pp. 335-336 (this issue).**[3]**R. B. Lakein,*A Gauss bound for a class of biquadratic fields*, J. Number Theory**1**(1969), 108–112. MR**0240073****[4]**Richard B. Lakein,*Computation of the ideal class group of certain complex quartic fields*, Math. Comp.**28**(1974), 839–846. MR**0374090**, 10.1090/S0025-5718-1974-0374090-1**[5]**Daniel Shanks,*On Gauss’s class number problems*, Math. Comp.**23**(1969), 151–163. MR**0262204**, 10.1090/S0025-5718-1969-0262204-1**[6]**D. SHANKS, "Review of Table: Class numbers of primes of the form ,"*Math. Comp.*, v. 23, 1969, pp. 213-214.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1975-0444605-4

Article copyright:
© Copyright 1975
American Mathematical Society