Search

Let $\Gamma \,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a natural number $5^{th}$ power-free. Let $k = \mathbb{Q}(\sqrt[5]{n}, \zeta_5)$, with $\zeta_5$ is a primitive $5^{th}$ root of unit, be the normal closure of $\Gamma$, and a pure metacyclic field of degree $20$ over $\mathbb{Q}$. When $n$ takes some particular forms, we show that $\Gamma$ admits a trivial $5$-class group and $5$ divides exactly the class number of $k$., Comment: 10 pages, 3 tables

Let $G$ be a $5$-group of maximal class and $\gamma_2(G) = [G, G]$ its derived group. Assume that the abelianization $G/\gamma_2(G)$ is of type $(5, 5)$ and the transfers $V_{H_1\to \gamma_2(G)}$ and $V_{H_2\to \gamma_2(G)}$ are trivial, where $H_1$ and $H_2$ are two maximal normal subgroups of $G$. Then $G$ is completely determined with the isomorphism class groups of maximal class. Moreover the group $G$ is realizable with some fields $k$, which is the normal closure of a pure quintic field., Comment: 9 pages, 1 figures, 1 table

Let $n$ be a $5^{th}$ power-free naturel number and $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field generated by a primitive $5^{th}$ root of unity $\zeta_5$. Then $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ is a pure metacyclic field of absolute degree $20$. In the case that $k$ possesses a $5$-class group $C_{k,5}$ of type $(5,5)$ and all the classes are ambiguous under the action of $Gal(k/k_0)$, the capitulation of $5$-ideal classes of $k$ in its unramified cyclic quintic extensions is determined., Comment: 12 pages

Let $\Gamma \,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer $5^{th}$ power-free, $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $\zeta_5$, and $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ the normal closure of $\Gamma$. Let $k_5^{(1)}$ be the Hilbert $5$-class field of $k$, $C_{k,5}$ the $5$-ideal classes group of $k$, and $C_{k,5}^{(\sigma)}$ the group of ambiguous classes under the action of $Gal(k/k_0)$ = $\langle\sigma\rangle$. When $C_{k,5}$ is of type $(5,5)$ and rank $C_{k,5}^{(\sigma)}\,=\,1$, we study the capitulation problem of the $5$-ideal classes of $C_{k,5}$ in the six intermediate extensions of $k_5^{(1)}/k$., Comment: 10 pages, 1 figures

In this paper, we study the length of the $2$-class field towers and the structure of the Galois groups $\mathrm{Gal}(\mathcal{L}(K_n)/K_n)$ of the maximal unramified $2$-extensions of the layers $K_n$ of the cyclotomic $\mathbb{Z}_2$-extension of some special Dirichlet fields. The capitulation problem is investigated too., Comment: 15 pages

Let $\Gamma \,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer, $5^{th}$ power-free. Let $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $\zeta_5$, and $k\,=\,\Gamma(\zeta_5)$ be the normal closure of $\Gamma$. Let $C_{k,5}$ be the $5$-component of the class group of k. The purpose of this paper is to determine generators of $C_{k,5}$, whenever it is of type $(5,5)$ and the rank of the group of ambiguous classes under the action of $Gal(k/k_0)\, =\,\langle \sigma\rangle$ is $1$., Comment: 19 pages, 3 tables

In this paper, we determine the 2-rank of the class group of certain classes of real cyclic quartic number fields. Precisely, we consider the case in which the quadratic subfield is Q(\sqrt{l}) with l=2 or a prime congruent to 1 mod 8.

In this paper, we investigate the 2-rank of the class group of some real cyclic quartic number fields. Precisely, we consider the case where the quadratic subfield is Q(\sqrt{l}) with l congruent to 5 modulo 8 is a prime.

In this paper, we investigate the unit groups, the $2$-class groups, the $2$-class field towers and the structures of the second $2$-class groups of some multiquadratic number fields of degree $8$ and $16$., Comment: Published online in Turk J Math. 18 pages

Let $k \,=\, \mathbb{Q}(\sqrt[5]{n},\zeta_5)$, where $n$ is a positive integer, $5^{th}$ power-free, whose $5-$class group is isomorphic to $\mathbb{Z}/5\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$. Let $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $\zeta_5$. Let $C_{k,5}^{(\sigma)}$ the group of the ambiguous classes under the action of $Gal(k/k_0)$ = $<\sigma>$. The aim of this paper is to determine all integers $n$ such that the group of ambiguous classes $C_{k,5}^{(\sigma)}$ has rank $1$ or $2$., Comment: 18 pages, 1 figure

Let $d$ be a square free integer and $L_d:=\mathbb{Q}(\zeta_{8},\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, $\mathrm{Cl}_2(L_d)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$., Comment: 14 pages

Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic $\mathbb{Z}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m, d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5\pmod 8$., Comment: 12 pages

Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The precise length ell5(M) of the 5-class tower of M is determined, when G(5,2)M is of order less than or equal to 5^5. Theoretical results are underpinned by the actual computation of all 83, respectively 93, cases in the range 0

Let $\mathit{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$, where $d>1$ is a cube-free positive integer, $\mathit{k}_0=\mathbb{Q}(\zeta_3)$ be the cyclotomic field containing a primitive cube root of unity $\zeta_3$, and $G=\operatorname{Gal}(\mathit{k}/\mathit{k}_0)$. The possible prime factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new phenomena concerning the chain $\Theta=(\theta_i)_{i\in\mathbb{Z}}$ of \textit{lattice minima} in the underlying pure cubic subfield $L=\mathbb{Q}(\sqrt[3]{d})$ of $\mathit{k}$. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals $(\alpha)\in\mathcal{P}_{\mathit{k}}^G/\mathcal{P}_{\mathit{k}_0}$ among the lattice minima $\Theta=(\theta_i)_{i\in\mathbb{Z}}$ of the underlying pure cubic field $L=\mathbb{Q}(\sqrt[3]{d})$, and to explain exceptional behavior of the chain $\Theta$ for certain radicands $d$ with impact on determining the principal factorization type of $L$ and $\mathit{k}$ by means of Voronoi's algorithm., Comment: 17 pages, 5 Tables

Let $d$ be an odd square-free integer and $\zeta_8$ a primitive $8$-th root of unity. The purpose of this paper is to investigate the rank of the $2$-class group of the fields $L_d=\mathbb{Q}(\zeta_8,\sqrt{d})$., Comment: 19 pages

Let $p\equiv 1\,(\mathrm{mod}\,3)$ be a prime and denote by $\zeta_3$ a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about $3$-class groups of pure cubic fields $L=\mathbb{Q}(\sqrt[3]{p})$ and of their normal closures $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$. The purpose of this paper is to prove Lemmermeyer's conjecture., Comment: Pure cubic fields, Galois closure, $3$-class groups, abelian type invariants

Let $N=K(\sqrt[3]{D})$ be a cubic Kummer extension of the cyclotomic field $K=\mathbb{Q}(\zeta_3)$, containing a primitive cube root of unity $\zeta_3$, with cube free integer radicand $D>1$. Denote by $f$ the conductor of the abelian extension $N/K$, and by $N^{\ast}$ the relative genus field of $N/K$. The aim of the present work is to find out all positive integers $D$ and conductors $f$ such that the genus group $\operatorname{Gal}\left(N^{\ast}/N\right)\cong \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ is elementary bicyclic., Comment: 22 pages, 2 figures, 8 tables

Let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$, with $d$ a cube-free positive integer. Let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. By the aid of genus theory, arithmetic proprieties of the pure cubic field $\mathbb{Q}(\sqrt[3]{d})$ and some results on the $3$-class group $C_{\mathrm{k},3}$, we are moving towards the determination of all integers $d$ such that $C_{\mathrm{k},3} \simeq \mathbb{Z}/9\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$., Comment: 10 pages

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(\zeta_3)$ with $d>1$ a cube-free integer and $\zeta_3$ a primitive third root of unity. Denote by $C_{k,3}^{(\sigma)}$ the $3$-group of ambiguous classes of the extension $k/k_0$ with relative group $G=\operatorname{Gal}(k/k_0)=\langle\sigma\rangle$. The aims of this paper are to characterize all extensions $k/k_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(\sigma)}$ 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., Comment: 24 pages, 7 tables, 3 figures

Let $\mathrm{k}=\mathbb{Q}\left(\sqrt[3]{p},\zeta_3\right)$, where $p$ is a prime number such that $p \equiv 1 \pmod 9$, and let $C_{\mathrm{k},3}$ be the $3$-component of the class group of $\mathrm{k}$. In \cite{GERTH3}, Frank Gerth III proves a conjecture made by Calegari and Emerton \cite{Cal-Emer} which gives necessary and sufficient conditions for $C_{\mathrm{k},3}$ to be of $\operatorname{rank}\,$ two. The purpose of the present work is to determine generators of $C_{\mathrm{k},3}$, whenever it is isomorphic to $\mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$., Comment: 16 pages, 2 tables

We study the capitulation of $2$-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields $k =Q(\sqrt{pq_1q_2}, i)$, where $i=\sqrt{-1}$ and $q_1\equiv q_2\equiv-p\equiv-1 \pmod 4$ are different primes. For each of the three quadratic extensions $K/k$ inside the absolute genus field $k^{(*)}$ of $k$, we compute the capitulation kernel of $K/k$. Then we deduce that each strongly ambiguous class of $k/Q(i)$ capitulates already in $k^{(*)}$., Comment: 17 pages, To be published by Acta Mathematica Vietnamica, 2016. arXiv admin note: substantial text overlap with arXiv:1507.00295, arXiv:1503.01992

We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\mathbf{k} =\mathbb{Q}(\sqrt{p_1p_2q}, i)$, where $i=\sqrt{-1}$ and $p_1\equiv p_2\equiv-q\equiv1 \pmod 4$ are different primes. For each of the three quadratic extensions $\mathbf{K}/\mathbf{k}$ inside the absolute genus field $\mathbf{k}^{(*)}$ of $\mathbf{k}$, we compute the capitulation kernel of $\mathbf{K}/\mathbf{k}$. Then we deduce that each strongly ambiguous class of $\mathbf{k}/\mathbb{Q}(i)$ capitulates already in $\mathbf{k}^{(*)}$, which is smaller than the relative genus field $(\mathbf{k}/\mathbb{Q}(i))^*$., Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1503.01992

Let $p_1\equiv p_2\equiv5 \pmod8$ be prime numbers such that $\left(\frac{p_1}{p_2}\right)=-1$. Let $\mathbb{L}=Q(\sqrt2, \sqrt{p_1p_2})$ Our goal is to resolve the equation $X^2-\varepsilon_2\varepsilon_{p_1p_2}\varepsilon_{2p_1p_2}=0$ in $\mathbb{L}$, where $\varepsilon_j$ are fundamental units of real quadratic subfields of $Q(\sqrt2, \sqrt{p_1p_2})$., Comment: in French

Let $p_1$ and $p_2$ be two primes such that $p_1\equiv p_2\equiv1 \pmod4$ and at least two of the three elements $\{(\frac{2}{p_1}), (\frac{2}{p_2}), (\frac{p_1}{p_2})\}$ are equal to -1. Put $i=\sqrt{-1}$, $d=2p_1p_2$ and $k =Q(\sqrt{d}, i)$. Let $k_2^{(1)}$ be the Hilbert 2-class field of $k$ and $k^{(*)}=Q(\sqrt{p_1},\sqrt{p_2},\sqrt 2, i)$ be its genus field. Let $C_{k,2}$ denote the 2-part of the class group of $k$. The unramified abelian extensions of $k$ are $K_1=k(\sqrt{p_1})$, $K_2=k(\sqrt{p_2})$, $K_3=k(\sqrt{2})$ and $k^{(*)}$. Our goal is to study the capitulation problem of the 2-classes of $k$ in these four extensions., Comment: 6 pages, in French in Workshop International "Th\'eorie des Nombres, Codes, Cryptographie et Syst\`emes de Communication", du 26 au 28 Avril 2012 01/2012

Let $p_1 \equiv p_2 \equiv5\pmod8$ be different primes. Put $i=\sqrt{-1}$ and $d=2p_1p_2$, then the bicyclic biquadratic field $k=Q(\sqrt{d}, \sqrt{-1})$ has an elementary abelian 2-class group of rank $3$. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group $\mathrm{Gal}(k_2^{(2)}/k)$ of the second Hilbert 2-class field $k_2^{(2)}$ of $k$. We study the capitulation problem of the 2-classes of $k$ in its seven unramified quadratic extensions $K_i$ and in its seven unramified bicyclic biquadratic extensions $L_i$.

We construct an infinite family of imaginary bicyclic biquadratic number fields $k$ with the 2-ranks of their 2-class groups are $\geq3$, whose strongly ambiguous classes of $k/Q(i)$ capitulate in the absolute genus field $k^{(*)}$, which is strictly included in the relative genus field $(k/Q(i))^*$ and we study the capitulation of the $2$-ideal classes of $k$ in its quadratic extensions included in $k^{(*)}$.

We study the capitulation of ideal classes in an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $k =Q(\sqrt{2pq}, i)$, where $i=\sqrt{-1}$ and $p\equiv -q\equiv1 \pmod 4$ are different primes. For each of the three quadratic extensions $K/k$ inside the absolute genus field $k^{(*)}$ of $k$, we compute the capitulation kernel of $K/k$. Then we deduce that each strongly ambiguous class of $k/Q(i)$ capitulates already in $k^{(*)}$, which is smaller than the relative genus field $\left(k/Q(i)\right)^*$.

Let $G$ be some metabelian $2$-group satisfying the condition $G/G'\simeq \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$. In this paper, we construct all the subgroups of $G$ of index $2$ or $4$, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem of the $2$-ideal classes of some fields $\mathbf{k}$ satisfying the condition $\mathrm{G}al(\mathbf{k}_2^{(2)}/\mathbf{k})\simeq G$, where $\mathbf{k}_2^{(2)}$ is the second Hilbert $2$-class field of $\mathbf{k}$., Comment: in Journal of Taibah University for Science (2015)

Let $p_1\equiv p_2\equiv -q\equiv1 \pmod4$ be different primes such that $\displaystyle\left(\frac{2}{p_1}\right)= \displaystyle\left(\frac{2}{p_2}\right)=\displaystyle\left(\frac{p_1}{q}\right)=\displaystyle\left(\frac{p_2}{q}\right)=-1$. Put $d=p_1p_2q$ and $i=\sqrt{-1}$, then the bicyclic biquadratic field ${k}=\mathbb{Q}(\sqrt{d},i)$ has an elementary abelian $2$-class group, $\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of ${k}$ in its fourteen unramified abelian extensions $\mathbb{K}_j$ and $\mathbb{L}_j$ within ${k}_2^{(1)}$, that is the Hilbert $2$-class field of ${k}$. We determine the nilpotency class, the coclass, generators and the structure of the metabelian Galois group $G=\mathrm{Gal}(\mathbb{L}/{k})$ of the second Hilbert 2-class field ${k}_2^{(2)}$ of ${k}$. Additionally, the abelian type invariants of the groups $\mathbf{C}l_2(\mathbb{K}_j)$ and $\mathbf{C}l_2(\mathbb{L}_j)$ and the length of the $2$-class tower of ${k}$ are given.

Let $p$ be a prime number such that $p\equiv 1$ mod $8$ and $i=\sqrt{-1}$. Let $\mathbf{k}=\mathbb{Q}(\sqrt{2p}, i)$, $\mathbf{k}_1^{(2)}$ be the Hilbert $2$-class field of $\mathbf{k}$, $\mathbf{k}_2^{(2)}$ be the Hilbert $2$-class field of $\mathbf{k}_1^{(2)}$ and $G=\mathrm{Gal}(\mathbf{k}_2^{(2)}/\mathbf{k})$ be the Galois group of $\mathbf{k}_2^{(2)}/\mathbf{k}$. Suppose that the $2$-part, $C_{\mathbf{k}, 2}$, of the class group of $\mathbf{k}$ is of type $(2, 4)$; then $\mathbf{k}_1^{(2)}$ contains six extensions $\mathbf{K_{i, j}}/\mathbf{k}$, $i=1, 2, 3$ and $j=2, 4$. Our goal is to study the problem of the capitulation of $2$-ideal classes of $\mathbf{K_{i, j}}$ and to determine the structure of $G$.

Quantum Cryptography or Quantum key distribution (QKD) is a technique that allows the secure distribution of a bit string, used as key in cryptographic protocols. When it was noted that quantum computers could break public key cryptosystems based on number theory extensive studies have been undertaken on QKD. Based on quantum mechanics, QKD offers unconditionally secure communication. Now, the progress of research in this field allows the anticipation of QKD to be available outside of laboratories within the next few years. Efforts are made to improve the performance and reliability of the implemented technologies. But several challenges remain despite this big progress. The task of how to test the apparatuses of QKD For example did not yet receive enough attention. These devises become complex and demand a big verification effort. In this paper we are interested in an approach based on the technique of probabilistic model checking for studying quantum information. Precisely, we use the PRISM tool to analyze the security of BB84 protocol and we are focused on the specific security property of eavesdropping detection. We show that this property is affected by the parameters of quantum channel and the power of eavesdropper., Comment: 12 Pages, IJNSA