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 $p$ 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...

Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k\left(\right(T\left)\right)$ and $A$ is an abelian $p$-adic Lie subgroup of ${\mathrm{Aut}}_k\left(K\right)$. In this paper we extend this equivalence to allow $\mathrm{Gal}(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups.

Let $k$ be a finite extension of ${\mathbb{Q}}_p$ and ${ℰ}_k$ be the set of the extensions of degree $p^2$ over $k$ whose normal closure is a $p$-extension. For a fixed discriminant, we show how many extensions there are in ${ℰ}_{{\mathbb{Q}}_p}$ with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in ${ℰ}_k$.

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 []. ...

1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0...