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Let $p=1+2^{e+1}q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e+1}$ (resp. order $d_{\phi }$ dividing q), and let ψₙ be an even character of conductor $p^{n+1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $Kₙ=\mathbb{Q}\left({\zeta }_{(p-1)pⁿ}\right)$. It is well known that the Bernoulli number $B_{1,\chi }$ is not zero, which is shown in an analytic way. In the extreme cases $d_{\phi }=1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T{r}_{n/1}\left(\xi B_{1,\chi }\right)\ne 0$ for any pⁿth root ξ...

Let $p$ be a prime number. A finite Galois extension $N/F$ of a number field $F$ with group $G$ has a normal $p$-integral basis ($p$-NIB for short) when ${𝒪}_N^{\prime }$ is free of rank one over the group ring ${𝒪}_F^{\prime }\left[G\right]$. Here, ${𝒪}_F^{\prime }={𝒪}_F[1/p]$ is the ring of $p$-integers of $F$. Let $m=p^e$ be a power of $p$ and $N/F$ a cyclic extension of degree $m$. When ${\zeta }_m\in F^{\times }$, we give a necessary and sufficient condition for $N/F$ to have a $p$-NIB (Theorem 3). When ${\zeta }_m\notin F^{\times }$ and $p\nmid [F\left({\zeta }_m\right):F]$, we show that $N/F$ has a $p$-NIB if and only if $N\left({\zeta }_m\right)/F\left({\zeta }_m\right)$ has a $p$-NIB (Theorem 1). When $p$ divides $[F\left({\zeta }_m\right):F]$, we show that this descent property...

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 Galois group...