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 $\left(k/Q(i)\right)^*$.
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 $k^{(*)}$.
For a Galois extension $K/F$ with $\text{char}(K)\neq 2$ and $\text{Gal}(K/F) \simeq \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, we determine the $\mathbb{F}_2[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times 2}$. Although there are an infinite number of (pairwise non-isomorphic) indecomposable $\mathbb{F}_2[\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}]$-modules, our decomposition includes at most $9$ indecomposable types. This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representatio…
This paper presents results on both the kernel and cokernel of the S-capitulation map C_{F,S}\ra C_{K,S}^{G} for arbitrary finite Galois extensions K/F (with Galois group G) and arbitrary finite sets of primes S of F (assumed to contain the archimedean primes in the number field case)
This paper introduces two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal. It generalizes techniques of Graves used to prove that the number field $\mathbb{Q}(\sqrt{2},\sqrt{35})$ has a non-principal Euclidean ideal.
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 t…
Let $k \,=\, \mathbb{Q}(\sqrt[5]{n},ζ_5)$, 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 $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $ζ_5$. Let $C_{k,5}^{(σ)}$ the group of the ambiguous classes under the action of $Gal(k/k_0)$ = $<σ>$. The aim of this paper is to determine all integers $n$ such that the group of ambiguous classes $C_{k,5}^{(σ)}$ has rank $1$ or $2$.
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 $k^{(*)}$.
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 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 sm…
Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(ζ_3)$ with $d>1$ a cube-free integer and $ζ_3$ a primitive third root of unity. Denote by $C_{k,3}^{(σ)}$ the $3$-group of ambiguous classes of the extension $k/k_0$ with relative group $G=\operatorname{Gal}(k/k_0)=\langleσ\rangle$. The aims of this paper are to characterize all extensions $k/k_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(σ)}$ 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…
Let $n$ be a $5^{th}$ power-free naturel number and $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field generated by a primitive $5^{th}$ root of unity $ζ_5$. Then $k\,=\,\mathbb{Q}(\sqrt[5]{n},ζ_5)$ 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.
Let $Γ\,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer $5^{th}$ power-free, $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $ζ_5$, and $k\,=\,\mathbb{Q}(\sqrt[5]{n},ζ_5)$ the normal closure of $Γ$. Let $k_5^{(1)}$ be the Hilbert $5$-class field of $k$, $C_{k,5}$ the $5$-ideal classes group of $k$, and $C_{k,5}^{(σ)}$ the group of ambiguous classes under the action of $Gal(k/k_0)$ = $\langleσ\rangle$. When $C_{k,5}$ is of type $(5,5)$ and rank $C_{k,5}^{(σ)}\,=\,1$, we study the capitulation problem of the $5…
Let $q$ be a power of a prime number $p$. Let $k=\mathbb{F}_{q}(t)$ be the rational function field with constant field $\mathbb{F}_{q}$. Let $K=k(α)$ be an Artin-Schreier extension of $k$. In this paper, we explicitly describe the ambiguous ideal classes and the genus field of $K$ . Using these results we study the $p$-part of the ideal class group of the integral closure of $\mathbb{F}_{q}[t]$ in $K$. And we also give an analogy of R$\acute{e}$dei-Reichardt's formulae for $K$.
For $L/K$ a finite Galois extension of number fields, the relative Pólya group $\Po(L/K)$ coincides with the group of strongly ambiguous ideal classes in $L/K$. In this paper, using a well known exact sequence related to $\Po(L/K)$, in the works of Brumer-Rosen and Zantema, we find short proofs for some classical results in the literatur. Then we define the ``Ostrowski quotient'' $\Ost(L/K)$ as the cokernel of the capitulation map into $\Po(L/K)$, and generalize some known results for $\Po(L/\mathbb{Q})$ to $\Ost(L/K)$.
For a cyclic Kummer extension $K$ of a rational function field $k$ is considered, via class field theory, the extended Hilbert class field $K_H^+$ of $K$ and the corresponding extended genus field $K_g^+$ of $K$ over $k$, along the lines of the definitions of R. Clement for such extensions of prime degree. We obtain $K_g^+$ explicitly. Also, we use cohomology to determine the number of ambiguous classes and obtain a reciprocity law for $K/k$. Finally, we present a necessary and sufficient condition for a prime of $K$ to decompose fully in $K_g^+$.
Let $Γ\,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer, $5^{th}$ power-free. Let $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $ζ_5$, and $k\,=\,Γ(ζ_5)$ be the normal closure of $Γ$. 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 $Gal(k/k_0)\, =\,\langle σ\rangle$ is $1$.
An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the algorithms for ideal manipulation. However, this is not the only option, as an ideal can also be represented by its generating set. Moreover, in certain quadratic orders of the ring of integral quaternions, every ideal can be represented by a single quaternion, called a pseudo generator. In this paper, we develop algorithms which allow us to manipulate ideals using s…
