# of Karim Belabas, Aïssa Derhem, and Daniel C. Mayer

 * On these pages, we present most recent results of our joint research, directly from the lab. * Basic bibliography: K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237 A. Derhem, Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Québec, 1988 D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58 * E-mail addresses: Karim.Belabas@math.u-psud.fr aderhem@yahoo.fr danielmayer@algebra.at *

 Pure Cubic Fields with 3-class rank 0 (2002/10/02) Dan (02/10/02): With the present communication I start a new series of expositions concerning the 3-class groups of pure cubic fields and their normal closures. Let K = Q( R1/3 ) be a pure cubic field with normalized radicand R = ab2, where a > b >= 1 are square-free coprime integers. The normal closure of K is the S3-field N = K( zeta ) with the primitive 3rd root of unity zeta = exp(2 pi i/3). N is cyclic cubic over its quadratic cyclotomic subfield k = Q( zeta ) with conductor f = 3ab, if R is incongruent 1,8 (mod 9) (field of Dedekind's 1st kind), and f = ab, if R is congruent 1,8 (mod 9) (field of Dedekind's 2nd kind). We denote by hK, rK, and SK the class number, the 3-class rank, and the 3-class group (Sylow 3-subgroup of the class group) of K, respectively. In 1969, Pierre Barrucand and Harvey Cohn [1] gave a necessary condition for the 3-class group SK of K to be trivial ( SK = 1 <==> rK = 0 <==> 3 doesn't divide hK ) in terms of possible prime decompositions of the radicand R. One year later, Taira Honda [2] proved (with the aid of Hilbert's norm residue symbols ( zeta, R / P ) of N | k for prime ideals P of k) that this condition is also sufficient for SK = 1. The conductors f and radicands R in Honda's criterion are summarized in the following table. The multiplicity of the conductor m = m(f), i. e., the number of non-isomorphic fields sharing f and forming a so called multiplet of pure cubics , is a function of u, v, and e: u = #{ q prime | q divides f, q congruent 1,8 (mod 9) } (free prime divisors), v = #{ q prime | q divides f, q incongruent 1,8 (mod 9) } (restrictive prime divisors), and e = v3(f) is the exact 3-exponent of f that can be used to characterize Dedekind's types 1A (e = 2), 1B (e = 1), and 2 (e = 0). Most of these fields occur as singulets (m = 1). But the fields with e = 2 (except for f = 9) arise in 3519 complete doublets (m = 2) of type (BETA,BETA) and 6060 pseudo singulets of type (BETA), where the companion radicand exeeds 106 in [5] ( 2*3519 + 6060 = 13098 ). If the Principal Factorization Type is BETA, two principal factors (p.f.) are listed, which are minimal, in general (except when q1 < q12 < q2). To permit a feeling how frequently these radicands occur I give the counts (#) in my recent most extensive table [5] of the 827600 pure cubic fields Q( R1/3 ) with R < 106 and all occurrences below 100 as examples.

f R e u v m conditions SK type p.f. # examples
9 3 2 0 1 1 1 GAMMA 1 R = 3 (unique)
q q 0 1 0 1 q = 8 (mod 9) 1 GAMMA 13099 R = 17,53,71,89
3q q 1 0 2 1 q = 2,5 (mod 9) 1 BETA 3,9 26167 R = 2,5,11,23,29,41,47,59,83
9q 3q,9q 2 0 2 2 q = 2,5 (mod 9) 1 BETA 3,9 13098 R = 6,12,15,33,45,69,87,99
q1q2 q1q2,q12q2 0 0 2 1 q1,q2 = 2,5 (mod 9) 1 BETA q1,q2 21520 R = 10,44,46,55,82

 The class number relation of Arnold Scholz [0] , hN = hk * hK2 * ui / 3, where hk = 1 for the special field k = Q( zeta ) and ui denotes the index ( UN : Uo ) in { 1, 3 } ("old unit" index) of the subgroup Uo generated by the units of all proper subfields of N in the unit group UN of N, shows that the class number hN of N is coprime to 3 if and only if hK is coprime to 3. Moreover, for the fields with hK coprime to 3, whose conductors f and radicands R are mentioned in the table above, we must necessarily have unit index ui = 3, i. e., Principal Factorization Type BETA or GAMMA. These facts were observed in 1970 by Barrucand and Cohn [3] . Honda's criterion for pure cubics with 3-rank rK = 0 is intimately connected with the 1970 results of Hideo Wada [4] , who considers general cubic Kummer extensions of k = Q( zeta ) of which the normal closures of pure cubics are special cases. Concerning pure cubics with positive 3-rank rK > 0 see my next communication .

 References: [0] Arnold Scholz, Idealklassen und Einheiten in kubischen Körpern, Monatsh. Math. Phys. 40 (1933), 211 - 222 [1] Pierre Barrucand and Harvey Cohn, A rational genus, class number divisibility, and unit theory for pure cubic fields, J. Number Theory 2 (1970), 7 - 21 [2] Taira Honda, Pure cubic fields whose class numbers are multiples of three, J. Number Theory 3 (1971), 7 - 12 [3] Pierre Barrucand and Harvey Cohn, Remarks on principal factors in a relative cubic field, J. Number Theory 3 (1971), 226 - 239 [4] Hideo Wada, On cubic Galois extensions of Q( (-3)1/2 ), Proc. Japan Acad. 46 (1970), 397 - 400, A table of fundamental units of purely cubic fields, Proc. Japan Acad. 46 (1970), 1135 - 1140 [5] Daniel C. Mayer, Principal Factorization Types of Multiplets of Pure Cubic Fields Q( R1/3 ) with R < 106, Univ. Graz, Computer Centre, 2002