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In this paper, we investigate the cyclicity of the 2-class group of the first Hilbert 2-class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let p1 ≡ p2 ≡-q ≡ 1(mod 4) be different prime integers. Put K = Q(√p1 p2 q), and denote by C,2 its 2-class group and by 2(1) (respectively 2(2)) its first (respectively second) Hilbert 2-class field. Then, we are interested in studying the metacyclicity of G = Gal( 2(2)/) and the cyclicity of Gal( 2(2)/ 2(1)) whenever the 4-rank of C,2 is 1.

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... Denote by H i (i = 1, 2, 3), the three maximal subgroups of G, then G is metacyclic if and only if d(H i ) ≤ 2 for each i = 1, 2, 3. Proof. If G is abelian the result follows easily, else see Proposition 2.1 in [3]. ...

... Proof. For the case δ = 1 see [3] and for the case δ = 2, the result is obtained proceeding similarly as in [3]. ...

... Proof. For the case δ = 1 see [3] and for the case δ = 2, the result is obtained proceeding similarly as in [3]. ...

In this paper, we establish a necessary and sufficient criterion for a finite metabelian 2-group G whose abelianized Gab is of type (2,2m), with m≥2, to be metacyclic. This criterion is based on the rank of the maximal subgroup of G which contains the three normal subgroups of G of index 4. Then, we apply this result to study the structure of the Galois group of the maximal unramified pro-2-extension of the cyclotomic Z2-extension of certain number fields. Illustration is given by some real quadratic fields.

... For (q 1 , q 2 , q 3 ) =(3,11, 43), we have 3 ≡ 11 ≡ 43 ≡ −1 (mod 4) 1 ) A(k 2 ) (4, 4) , A(k 1 ( √ q i )) A(k 2 ( √ q i )) (2, 4) for all i ∈ {1, 2, 3}By Theorems 4.3 and 4.8, G (4, 4).2. For(q 1 , q 2 , q 3 ) = (3, 11, 19), we have 3 ≡ 11 ≡ 19 ≡ −1 (mod 4) , A(k 2 ) (8, 8) and A(k 3 ) A(k 4 ) (16, 16), A(k 1 ( √ q i )) (2, 4) and A(k 2 ( √ q i )) (4, 8), for all i ∈ {1, 2, 3} By Theorems 4.3 and 4.8, G (16,16). ...

For a number field $ k$, we consider the Galois group $G=\mathrm{Gal}(\mathcal{L}( k_{\infty})/ k_{\infty})$ of the maximal unramified pro-$2$-extension of the cyclotomic $\mathbb{Z}_2$-extension $ k_{\infty}$ of $ k$. In terms of transfer, we establish a necessary and sufficient condition for a $2$-group to be abelian or metacyclic non-abelian whenever its abelianization is of type $(2^n, 2^m)$, with $n\geq2$ and $m\geq2$. Then we apply this result to construct an infinite family of real quadratic fields for which $G$ is an abelian pro-$2$-group of rank $2$.

Let k be a real quadratic number field with 2-class group isomorphic to , , , and let be the Hilbert 2-class field of k. We give complete criteria for to be cyclic when either , the discriminant of k, is divisible by only positive prime discriminants, or when the 2-class number of is greater than 2, and partial criteria for to be elementary cyclic when is divisible by a negative prime discriminant.

Let k be an imaginary quadratic number field and k1 the Hilbert 2-class field of k. We give a characterization of those k with Cl2(k) ≃ (2, 2m) such that Cl2(k1) has 2 generators.

Let k be an imaginary quadratic number field. Let k 1 denote the Hilbert 2-class field of k. We characterize those k whose 2-class fields have cyclic 2-class group.

On the rank of the 2 2 -class group of Q ( m , d ) Q({\sqrt {m}},{\sqrt {d}}) . Let d d be a square-free positive integer and p p be a prime such that p ≡ 1 ( m o d 4 ) p\equiv 1\,(mod\, 4) . We set K = Q ( m , d ) K = Q({\sqrt {m}},{\sqrt {d}}) , where m = 2 m=2 or m = p m=p . In this paper, we determine the rank of the 2 2 -class group of K K .
Résumé. Soit K = Q ( m , d ) K = Q({\sqrt {m}},{\sqrt {d}}) , un corps biquadratique où m = 2 m=2 ou bien un premier p ≡ 1 ( m o d 4 ) p\equiv 1\,(mod\,4) et d d étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du 2 2 -groupe de classes de K K .

Sur les d'idéaux dans les extensions cycliques relatives de degré premier Annales de l'institut Fourier, tome 23, n o 4 (1973), p. 1-44. <http://www.numdam.org/item?id=AIF_1973__23_4_1_0> © Annales de l'institut Fourier, 1973, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/), implique l'accord avec les conditions gé-nérales d'utilisation (http://www.numdam.org/legal.php). Toute utilisa-tion commerciale ou impression systématique est constitutive d'une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Let K = ℚ(√1, √2), where d 1 and d2 are positive square-free integers such that (d1, d2) = 1. Let K2(1) be the Hilbert 2-class field of K. Let K2(2) be the Hubert 2-class field of K2(1) and K(*) the genus field of K. We suppose that K2(1) ≠ K(*) and GalK 2(1)/K) ≃ ℤ/2ℤ × ℤ/2ℤ. We study the capitulation problem of the 2-ideal classes of K in the sub-extensions of K2(1)/K and we determine the structure of Gal(K 2(2)/K).

We classify all complex quadratic number fields with 2-class group of type (2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2. These fields all have 2-class field tower of length 2. We still don't know examples of fields with 2-class field tower of length 3, but the smallest candidate is the field with discriminant -1015.

We characterize all finite metabelian 2-groups G whose abelianizations Gab are of type (2,2ⁿ), with n≥2, and for which their commutator subgroups G′ have rank=2. This is given in terms of the order of the abelianizations of the maximal subgroups and the structure of the abelianizations of those normal subgroups of index 4 in G. We then translate these group theoretic properties to give a characterization of number fields k with 2-class group Cl2(k)≃(2,2ⁿ), n≥2, such that the rank of Cl2(k¹)=2 where k¹ is the Hilbert 2-class field of k. In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.

On the rank of the 2-class group of Q(√m, √d). Let d be a square-free positive integer and p be a prime such that p = 1 (mod4). We set K = Q(√m, √d), where m = 2 or m = p. In this paper, we determine the rank of the 2-class group of K.

Letkbe an imaginary quadratic number field. Letk1denote the Hilbert 2-class field ofk. We characterize thosekwhose 2-class fields have cyclic 2-class group.

Let k be a number field with Sk, the Sylow 2-subgroup of its ideal class group, isomorphic to the four-group. Then either the class number of the Hilbert class field to k is odd, or there is a unique nonabelian unramified extension L of k of degree 8. The galois group is then the dihedral or quaternion group of order 8, and the occurrence of each is characterized in terms of Hilbert's theorem 94. In the case , m a positive square-free integer, we obtain this characterization in terms of arithmetic properties of the integer m.

Burnside ((1), p. 241) has proved the following theorem:
If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a non-Abelian group of order p ³ cannot be the derived group of a p-group .

We determine all real quadratic number fields with 2-class field tower of length at most 1.

Number Fields with 2-class Number Isomorphic to (2, 2 m ), preprint

- E Benjamin
- C Snyder

E. Benjamin and C. Snyder, Number Fields with 2-class Number Isomorphic
to (2, 2 m ), preprint, 1994.

Infinite Class Field Towers of Quadratic Fields

- E S Golod
- I R Shararevich

E. S. Golod, I. R. Shararevich, Infinite Class Field Towers of Quadratic
Fields, Izv. Akad. Nauk. SSSR 28 (1964), 273-276, (also see AMS Transl.
48 (1965), 91-102).

- A Azizi
- A Zekhnini
- M Taous

A. Azizi, A. Zekhnini and M. Taous, Capitulation in Abelian extensions of some fields
Q(
√ p 1 p 2 q, i), AIP Conf. Proc. 1705(1) (2016) 1-8; arXiv:1507.00295.

The Groups of Order 2 n (n ≤ 6

- M Hall
- J K Senior

M. Hall and J. K. Senior, The Groups of Order 2 n (n ≤ 6) (Macmillan, New York
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Über die Löbarkeit der Gleichung t 2 − Du 2 = −4

- A Scholz

A. Scholz,Über die Löbarkeit der Gleichung t 2 − Du 2 = −4, Math. Z. 39 (1934)
95-111.