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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The simplest cubic fields


Author: Daniel Shanks
Journal: Math. Comp. 28 (1974), 1137-1152
MSC: Primary 12A50; Secondary 12A30
MathSciNet review: 0352049
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Abstract: The cyclic cubic fields generated by $ {x^3} = a{x^2} + (a + 3)x + 1$ are studied in detail. The regulators are relatively small and are known at once. The class numbers are always of the form $ {A^2} + 3{B^2}$, are relatively large and easy to compute. The class groups are usually easy to determine since one has the theorem that if m is divisible only by $ {\text{primes}} \equiv 2\pmod 3$, then the m-rank of the class group is even. Fields with different 3-ranks are treated separately.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1974-0352049-8
PII: S 0025-5718(1974)0352049-8
Article copyright: © Copyright 1974 American Mathematical Society