0. Introduction. Since ℤ is a principal ideal domain, every finitely generated torsion-free ℤ-module has a finite ℤ-basis; in particular, any fractional ideal in a number field has an "integral basis". However, if K is an arbitrary number field the ring of integers, A, of K is a Dedekind domain but not necessarily a principal ideal domain. If L/K is a finite extension of number fields, then the fractional ideals of L are finitely generated and torsion-free (or, equivalently, finitely generated and...

Let $L$ be a finite abelian extension of $\mathbb{Q}$, with ${𝒪}_L$ the ring of algebraic integers of $L$. We investigate the Galois structure of the unique fractional ${𝒪}_L$ -ideal which (if it exists) is unimodular with respect to the trace form of $L/\mathbb{Q}$.

Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of $\mathrm{Gal}(\overline{E}/E)$ satisfying a selfduality condition “of type $\mathrm{U}\left(3\right)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb{Q}]$. As important steps, and in any rank, we prove that any first order...

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 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). The case...

This paper provides a complete catalog of the break numbers that occur in the ramification filtration of fully and thus wildly ramified quaternion extensions of dyadic number fields which contain $\sqrt{-1}$ (along with some partial results for the more general case). This catalog depends upon the refined ramification filtration, which as defined in [2] is associated with the biquadratic subfield. Moreover we find that quaternion counter-examples to the conclusion of the Hasse-Arf Theorem are extremely rare...

A $p$-adic version of Stark’s Conjecture at $s=1$ is attributed to J.-P. Serre and stated (faultily) in Tate’s book on the Conjecture. Building instead on our previous paper (and work of Rubin) on the complex abelian case, we give a new approach to such a conjecture for real ray-class extensions of totally real number fields. We study the coherence of our $p$-adic conjecture and then formulate some integral refinements, both alone and in combination with its complex analogue. A ‘Weak Combined Refined’ version...