3-rank of ambiguous class groups of cubic Kummer extensions
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3-rank of ambiguous class groups of cubic Kummer extensions S. Aouissi2 · D. C. Mayer1 · M. C. Ismaili2 · M. Talbi3 · A. Azizi2
© Akadémiai Kiadó, Budapest, Hungary 2020
Abstract √ Let k = k0 ( 3 d) be a cubic Kummer extension of k0 = Q(ζ3 ) with d > 1 a cube-free integer (σ ) and ζ3 a primitive third root of unity. Denote by Ck,3 the 3-group of ambiguous classes of the extension k/k0 with relative group G = Gal(k/k0 ) = σ . The aims of this paper are to (σ ) characterize all extensions k/k0 with cyclic 3-group of ambiguous classes Ck,3 of order 3, to investigate the multiplicity m( f ) of the conductors f of these abelian extensions k/k0 , and to classify the fields k according to the cohomology of their unit groups E k as Galois modules over G. The techniques employed for reaching these goals are relative 3-genus fields, Hilbert norm residue symbols, quadratic 3-ring class groups modulo f , the Herbrand quotient of E k , and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results. Keywords Pure cubic fields · Cubic Kummer extensions · 3-group of ambiguous ideal classes · 3-rank · Hilbert 3-class field · Relative 3-genus field · Multiplicity of conductors · Galois cohomology of unit groups · Principal factorization types
Dedicated to the memory of Frank Emmett Gerth III.
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S. Aouissi [email protected] D. C. Mayer [email protected] http://www.algebra.at M. C. Ismaili [email protected] M. Talbi [email protected] A. Azizi [email protected]
1
Graz, Austria
2
Department of Mathematics, Mohammed first University, 717 Mohammed 6 street, 60000 Oujda, Morocco
3
Regional Center of Professions of Education and Training, 60000 Oujda, Morocco
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S. Aouissi et al.
Mathematics Subject Classification 11R11 · 11R16 · 11R20 · 11R27 · 11R29 · 11R37
1 Introduction √ Let d > 1 be a cube-free integer and k = Q( 3 d, ζ3 ) be a cubic Kummer extension of the cyclotomic field k0 = Q(ζ3 ). Denote by f the conductor of the abelian extension k/k0 , by (σ ) m = m( f ) its multiplicity, and by Ck,3 the 3-group of ambiguous ideal classes of k/k0 . Let ∗ ∗ k = (k/k0 ) be the maximal abelian extension of k0 contained in the Hilbert 3-class field k1 of k, which is called the relative 3-genus field of k/k0 (cf. [14, § 2, p. VII-3]). We consider the problem of finding the radicands d and conductors f of all pure cubic fields √ L = Q( 3 d) for which the Galois group Gal(k ∗ /k) is non-trivial cyclic. The present work gives the complete solution of this problem by characterizing all cubic Kummer extensions (σ ) k/k0 with cyclic 3-group of ambiguous ideal classes Ck,3 of order 3. In fact, we prove the following Main Theorem: √ (σ ) Theorem 1.1 Let k = Q( 3 d, ζ3 ), where d > 1 is a cube-free integer, and Ck,3 be the 3(σ )
group of ambiguous ideal classes of k/Q(ζ3 ). Then, rank (Ck,3 ) = 1 if and only if the integer d can be written in one of the following forms: ⎧ e1 p with p1 ≡ 1 (mod 3), ⎪ ⎪ ⎪ 3e1p e1 ⎪ with p1 ≡ 4 or 7 (mod 9), ⎪ ⎪ ⎪ e1 1 f
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