Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $\left(H_{p^n}^{\prime }\right)$ when any abelian extension $N/F$ of exponent dividing $p^n$ has a normal integral basis with respect to the ring of $p$-integers. We also say that $F$ satisfies $\left(H_{p^{\infty }}^{\prime }\right)$ when it satisfies $\left(H_{p^n}^{\prime }\right)$ for all $n\ge 1$. It is known that the rationals $\mathbb{Q}$ satisfy $\left(H_{p^{\infty }}^{\prime }\right)$ for all prime numbers $p$. In this paper, we give a simple condition for a number field $F$ to satisfy $\left(H_{p^n}^{\prime }\right)$ in terms of the ideal class group of $K=F\left({\zeta }_{p^n}\right)$ and a “Stickelberger ideal” associated to the...

We show how K. Hensel could have extended Wilson’s theorem from $\mathbf{Z}$ to the ring of integers $𝔬$ in a number field, to find the product of all invertible elements of a finite quotient of $𝔬$.

Let $k$ be the algebraic closure of ${𝔽}_p$ and $K$ be the local field of formal power series with coefficients in $k$. The aim of this paper is the description of the set ${𝒴}_n$ of conjugacy classes of series of order $p^n$ for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic $p$ which are invertible and of finite order $p^n$ for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series...

Let $p$ be a prime number, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two primes $𝔭$ and $\overline{𝔭}$. Let $k_{\infty }$ be the unique ${\mathbb{Z}}_p$-extension of $k$ which is unramified outside of $𝔭$, and let $K_{\infty }$ be a finite extension of $k_{\infty }$, abelian over $k$. Let ${𝒰}_{\infty }/{𝒞}_{\infty }$ be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of ${𝒰}_{\infty }/{𝒞}_{\infty }$ are finite. Our approach uses distributions and the $p$-adic $\mathrm{L}$-function, as defined in []. ...

Let K, L be algebraic number fields with K ⊆ L, and $O_K$, $O_L$ their respective rings of integers. We consider the trace map $T=T_{L/K}:L\to K$ and the $O_K$-ideal $T\left(O_L\right)\subseteq O_K$. By I(L/K) we denote the group indexof $T\left(O_L\right)$ in $O_K$ (i.e., the norm of $T\left(O_L\right)$ over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of $T\left(O_L\right)$ (Theorem 1)....

Let $G\cong C_{n_1}\oplus ...\oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1·...·w_n$ is a sequence of integers, all but at most one relatively prime to $\left|G\right|$, and $S$ is a sequence over $G$ with $\left|S\right|\ge \left|W\right|+\left|G\right|-1\ge \left|G\right|+1$, the maximum multiplicity of $S$ at most $\left|W\right|$, and $\sigma \left(W\right)\equiv 0mod\left|G\right|$, then there exists a nontrivial subgroup $H$ such that every element $g\in H$ can be represented as a weighted subsequence sum of the form $g=\underset{i=1}{\sum ^n}{w}_i{s}_i$, with $s_1·...·s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the...