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Let k = k 0 ( 3 √ d) be a cubic Kummer extension of k 0 = Q(ζ 3 ) with d > 1 a cubefree integer and ζ 3 a primitive third root of unity. Denote by C (σ) k,3 the 3-group of ambiguous classes of the extension k/k 0 with relative group G = Gal(k/k 0 ) = σ . The aims of this paper are to characterize all extensions k/k 0 with cyclic 3-group of ambiguous classes C (σ)k,3 of order 3, to investigate the multiplicity m(f ) of the conductors f of these abelian extensions k/k 0 , 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.

SUR LE RANG DU 2-GROUPE DE CLASSES DEABDELMALEK AZIZI AND ALI MOUHIB Abstract. On the rank of the 2-class group of Q(Let d be a square-free positive integer and p be a prime such that p ≡ 1 (mod 4). We, where m = 2 or m = p. In this paper, we determine the rank of the 2-class group of K., un corps biquadratique où m = 2 ou bien un premier p ≡ 1 (mod 4) et détant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du 2-groupe de classes de K.

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