Related papers: On totally real Hilbert-Speiser Fields of type C_p

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_\mathfrak{p}$ acting on $(\mathrm{mod}\, p^m)$ Katz Hilbert modular classes which agrees with the…

We give a necessary and sufficient condition for a prime to be an integer group determinant for an arbitrary abelian $p$-group of the form ${\rm C}_{p} \times H$, where ${\rm C}_{p}$ is the cyclic group of order $p$. Also, we show that…

Let $G$ be a finitely generated abelian-by-finite group and $k$ a field of characteristic $p\ge 0$. The Euler class $[k_G]$ of $G$ over $k$ is the class of the trivial $kG$-module in the Grothendieck group $G_0(kG)$. We show that $[k_G]$…

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the…

Let L/K be an extension of number fields where L/\Q is abelian. We define such an extension to be Leopoldt if the ring of integers O_L of L is free over the associated order A_L/K. Furthermore we define an abelian number field K to be…

There is no systematic general procedure by which isomorphism classes of Hopf algebras that are extensions of $\k F$ by ${\k}^G$ can be found. We develop the general procedure for classification of isomorphism classes of Hopf algebras which…

Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ be a primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ is a pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$.…

Using the description of dominions in the variety of nilpotent groups of class at most two, we give a characterization of which groups are absolutely closed in this variety. We use the general result to derive an easier characterization for…

We examine the phenomenon of capitulation of the $p$-class group $H_K$ of a real number field $K$ in totally ramified cyclic p-extensions $L/K$ of degree $p^N$. Using an elementary property of the algebraic norm $\nu_{L/K}$, we show that…

Let $F$ be a number field with ring of integers $O_F$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl(O_FG)$ of $O_FG$ that involves applying the work…

Let p be an odd prime. The lattice of all normal subgroups and the terms of the lower and upper central series are determined for all metabelian p-groups with generator rank d=2 having abelianization of type (p,p) and minimal defect of…

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring $\mathbb…

Let $k$ be a rational congruence function field and consider an arbitrary finite separable extension $K/k$. If for each prime in $k$ ramified in $K$ we have that at least one ramification index is not divided by the characteristic of $K$,…

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…

We introduce the abelian class group C_{ab}(G) of a reductive group scheme G over a ring A of arithmetical interest and study some of its properties. In particular, we show that if the fraction field of A is a global field without real…

For each odd prime $p$, we prove the existence of infinitely many real quadratic fields which are $p$-rational. Explicit imaginary and real bi-quadratic $p$-rational fields are also given for each prime $p$. Using a recent method developed…

Let $(K, v)$ be a Henselian valued field satisfying the following conditions, for a given prime number $p$: (i) central division $K$-algebras of (finite) $p$-primary dimensions have Schur indices equal to their exponents; (ii) the value…

Let $K$ be a number field, and let $G\subset K^\times$ be a finitely generated subgroup. Fix some prime number $\ell$, and consider the set of primes $\mathfrak{p}$ of $K$ satisfying the following property: the reduction of $G$ modulo…

Let $K$ be a number field and $f\in K[X]$ an irreducible monic polynomial with coefficients in $O_K$, the ring of integers of $K$. We aim to enounce an effective criterion, in terms of the Galois group of $f$ over $K$ and a linear…

We say a tame Galois field extension $L/K$ with Galois group $G$ has trivial Galois module structure if the rings of integers have the property that $\Cal{O}_{L}$ is a free $\Cal{O}_{K}[G]$-module. The work of Greither, Replogle, Rubin, and…