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On these pages, we present most recent results of our joint research, directly from the lab. 
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Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 12131237 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), 831847 and S55S58 
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Email addresses:
Karim.Belabas@math.upsud.fr aderhem@yahoo.fr danielmayer@algebra.at 
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Pure Cubic Fields with 3class rank 0 (2002/10/02) 
Dan (02/10/02):
With the present communication I start a new series of expositions concerning the 3class groups of pure cubic fields and their normal closures. Let K = Q( R^{1/3} ) be a pure cubic field with normalized radicand R = ab^{2}, where a > b >= 1 are squarefree coprime integers. The normal closure of K is the S_{3}field N = K( zeta ) with the primitive 3^{rd} 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 1^{st} kind), and f = ab, if R is congruent 1,8 (mod 9) (field of Dedekind's 2^{nd} kind). We denote by h_{K}, r_{K}, and S_{K} the class number, the 3class rank, and the 3class group (Sylow 3subgroup of the class group) of K, respectively. In 1969, Pierre Barrucand and Harvey Cohn [1] gave a necessary condition for the 3class group S_{K} of K to be trivial ( S_{K} = 1 <==> r_{K} = 0 <==> 3 doesn't divide h_{K} ) 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 S_{K} = 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 nonisomorphic 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 = v_{3}(f) is the exact 3exponent 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 10^{6} 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 q_{1} < q_{1}^{2} < q_{2}). 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( R^{1/3} ) with R < 10^{6} and all occurrences below 100 as examples. 

The class number relation of Arnold Scholz
[0]
,
h_{N} = h_{k} * h_{K}^{2} * ui / 3, where h_{k} = 1 for the special field k = Q( zeta ) and ui denotes the index ( U_{N} : U_{o} ) in { 1, 3 } ("old unit" index) of the subgroup U_{o} generated by the units of all proper subfields of N in the unit group U_{N} of N, shows that the class number h_{N} of N is coprime to 3 if and only if h_{K} is coprime to 3. Moreover, for the fields with h_{K} 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 3rank r_{K} = 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 3rank r_{K} > 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( R^{1/3} ) with R < 10^{6}, Univ. Graz, Computer Centre, 2002 
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