相关论文: Extensions of group schemes of $\mu$-type by a con…

We investigate the Galois group $G_S(p)$ of the maximal $p$-extension unramified outside a finite $S$ of primes of a number field in the (tame) case, when no prime dividing $p$ is in $S$. We show that the cohomology of $G_S(p)$ is 'often'…

We investigate the Galois group G_S(p) of the maximal p-extension unramified outside a finite set S of primes of a number field in the (mixed) case, when there are primes dividing p inside and outside S. We show that the cohomology of…

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete…

Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K = Q(zeta) be the p-cyclotomic field and let O_K be the ring of integers of K. Let pi be the prime ideal of K lying over p. An integer B \in O_K is…

Let $p$ be an irregular prime. Let $K=\Q(\zeta)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$.…

Every finite flat finitely presented group scheme G of square free order over a scheme S can be written as an extension of a finite etale S-group scheme G" by a commutative finite flat finitely presented S-group scheme G' that is a direct…

Let $G$ be a finite $p$-group. We construct a $G$-extension $K/k$ of number fields such that the $p$-adic completion of the unit group of $K$ has a prescribed $\mathbb{Z}_p[G]$-module structure, up to free direct summands.

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O_L is free as an O_K[G]-module. Let C_p denote…

For a prime ideal $\mathfrak{P}$ of the ring of integers of a number field $K$, we give a general definition of $\mathfrak{P}$-adic continued fraction, which also includes classical definitions of continued fractions in the field of…

In this work, we show that given a finite p-group G, a number field K having a trivial p-class group Cl K , and a finite set of primes S of K, there exists a finite extension F/K such that the S-split p-Hilbert class field tower L S p (F )…

For the ring R of integers of a ramified extension of the field of p-adic numbers and a cyclic group G of prime order p we study the extensions of the additive groups of R-representations modules of G by the group G.

We give a definition of associative schemes, schemes of associative rings, over a field $k,$ using the definition of completion of an associative $k$-algebra in a finite set of simple modules. We start by giving a weaker but sufficient…

Let $G$ be a finite group, and let $\mathbf{K}_p$ denote the completion at $p$ of the complex $K$-theory spectrum. $\mathbf{K}_p$ is a commutative ring spectrum that in some ways is very similar to the usual ring $\mathbf{Z}_p$ of $p$-adic…

There is a known analogy between growth questions for class groups and for Selmer groups. If $p$ is a prime, then the $p$-torsion of the ideal class group grows unboundedly in $\mathbb{Z}/p\mathbb{Z}$-extensions of a fixed number field $K$,…

We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes…

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 $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the…

Let $p$ be an odd prime number. For a degree $p$ extension of $p$-adic fields $L/K$, we give a complete characterization of the condition for the ring of integers $\mathcal{O}_L$ to be free as a module over its associated order in the…

For given rational prime number $p$ consider the tower of finite extensions of fields $K_0/\mathbb{Q}_p,$ $K/K_0, L/K, M/L$, where $K/K_0$ is unramified and $M/L$ is a Galois extension with Galois group $G$. Suppose one dimensional Honda…

We say that $M$ and $S$ form a \textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while $0$ has no such representation. The splitting is called {\it…