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.

Let [Formula: see text] with [Formula: see text] a cube-free positive integer. Let [Formula: see text] be the 3-class group of k. With the aid of genus theory, arithmetic properties of the pure cubic field [Formula: see text] and some results on the 3-class group [Formula: see text], we determine all integers [Formula: see text] such that [Formula: see text].

Let p ≡ 1 (mod 3) be a prime and denote by ζ 3 a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about 3-class groups of pure cubic fields L = Q( 3 √ p) and of their normal closures k = Q( 3 √ p, ζ 3 ). The purpose of this paper is to prove Lemmermeyer's conjecture.

Let be k=Q(\sqrt[3]{p},\zeta_3), where p is a prime number such that p \equiv 1 (mod  9), and let C_{k,3} the 3-component of the class group of k. In his work [7], Frank Gerth III proves a conjecture made by Calegari and Emerton which gives a necessary and sufficient conditions for C_{k,3} to be of rank two. The present work display a consideration steps towards determination of generators of C_{k,3}, when C_{k,3} is isomorphic to Z/9Z \times Z/3Z.