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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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Article copyright: © Copyright 1974 American Mathematical Society