Class Numbers of the Simplest Cubic Fields

Using the "simplest cubic fields" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by n, for any n. Using 2-descents on elliptic curves, we obtain precise information on the 2-Sylow subgroups of the class groups of these fields. A theorem of H. Heilbronn associates a set of quartic fields to the class group. We show how to obtain these fields via these elliptic curves. In [10], D. Shanks discussed a family of cyclic cubic fields and showed that they could be regarded as the cubic analogues of the real quadratic fields Q(\a2 + 4). These fields had previously appeared in the work of H. Cohn [4], who used them to produce cubic fields of even class number. Later, they appeared in the work of K. Uchida [12], who showed that for each n there are infinitely many cubic fields with class number divisible by n. In the following we first give another proof of Uchida's result and extend the techniques to handle some new cases. In the second part of the paper we study the relationship between elliptic curves and the 2-part of the class group, interpreting and extending the work of Cohn. 1. The Simplest Cubic Fields. Let m > 0 be an integer such that m & 3 mod 9. Let K be the cubic field defined by the irreducible (over Q) polynomial f(X) = X3 A mX2 -(mA 3)X + 1. The discriminant of f(X) is D2 = (m2 + 3m + 9)2 (note that m # 3 mod9 implies D # 0 mod27). Let p be the negative root of f(X). Then p' = 1/(1 p) and p" = 1 1/p are the other two roots, so K = Q(p) is a cyclic cubic field. Note that p, p', p" are units; in fact, p, p' are independent, hence generate a subgroup of finite index in the full group of units of K. Since -m2 < p < -m 1 < 0 < p' < 1 < p" < 2, it follows easily that all 8 combinations of signs may be obtained from units and their conjugates; hence, every totally positive unit is a square and the narrow and wide class numbers are equal. Received January 3, 1986. 1980 Mathematics Subject Classification. Primary 12A30, 12A50, 14K.07. "Research partially supported by NSF and the Max Planck Institut, Bonn. 371 ©1987 American Mathematical Society 0025-5718/87 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 372 LAWRENCE C. WASHINGTON Let a = —1 A p p2. Then a ( í Q) is a root of