Let $K$ be a finite extension of ${\mathbb{Q}}_p$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma$ and ramification index $p^n$. We give a criterion in terms of the ramification numbers $t_i$ for a fractional ideal ${𝔓}^h$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄(K\Gamma ;{𝔓}^h)$. In particular, if $t_n-[t_n/p]\<p^{n-1}e$ then the inverse different can be free over its associated order only when $t_i\equiv -1$ (mod $p^n$) for all $i$. We give three consequences of this. Firstly, if $𝔄(K\Gamma ;S)$ is a Hopf order and...

For $L/K$, any totally ramified cyclic extension of degree $p^2$ of local fields which are finite extensions of the field of $p$-adic numbers, we describe the ${\mathbb{Z}}_p\left[\mathrm{Gal}\right(L/K\left)\right]$-module structure of each fractional ideal of $L$ explicitly in terms of the $4p+1$ indecomposable ${\mathbb{Z}}_p\left[\mathrm{Gal}\right(L/K\left)\right]$-modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.

Let $L/K$ be an extension of algebraic number fields, where $L$ is abelian over $\mathbb{Q}$. In this paper we give an explicit description of the associated order ${𝒜}_{L/K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_L$, the ring of integers of $L$, is then isomorphic to ${𝒜}_{L/K}$. This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that ${𝒜}_{L/K}$ is the maximal order if $L/K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to ${\mathbb{Q}}^{\left(m^{\text{'}}\right)}/K$, where $m^{\text{'}}$ is the conductor...

We introduce a notion of “Galois closure” for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an $S_n$ degree $n$ extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions.

Let ${\mathbb{Z}}_d\wr$m$bethegeneralizedalternatinggroup.Weprovethatalldoublecoversof$ℤd ≀ m$occurasGaloisgroupsoveranyalgebraicnumberfield.WefurtherrealizesomeofthesedoublecoversastheGaloisgroupsofregularextensionsof\mathbb{Q}\left(T\right).Ifdisoddandm>7,theneverycentralextensionof$ℤd ≀ m$occursastheGaloisgroupofaregularextensionof\mathbb{Q}\left(T\right).Wefurtherimprovesomeofourearlierresultsconcerningdoublecoversofthegeneralizedsymmetricgroup$ℤd ≀ m$.$

Let $F$ be a number field with ring of integers $o_F$. For a fixed prime number $p$ and $i\ge 2$ the étale wild kernels $W{K}_{2i-2}^{\acute{\mathrm{e}}\mathrm{t}}\left(F\right)$ are defined as kernels of certain localization maps on the $i$-fold twist of the $p$-adic étale cohomology groups of $\mathrm{spec}{o}_F[\frac{1}{p}]$. These groups are finite and coincide for $i=2$ with the $p$-part of the classical wild kernel $W{K}_2\left(F\right)$. They play a role similar to the $p$-part of the $p$-class group of $F$. For class groups, Galois co-descent in a cyclic extension $L/F$ is described by the ambiguous class formula given...

The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_n$, can be embedded in any central extension of $A_n$ if and only if $n\equiv 0\left(mod8\right)$, or $n\equiv 2\left(mod8\right)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$, every central extension of $A_n$ occurs as a Galois group over $\mathbf{Q}$.

The main results of this paper may be loosely stated as follows. Theorem.— Let $N$ and $N^{\text{'}}$ be sums of Galois algebras with group $\Gamma$ over algebraic number fields. Suppose that $N$ and $N^{\text{'}}$ have the same dimension $\mathbb{Q}$ and that they are identical at their wildly ramified primes. Then (writing ${𝒪}_N$ for the maximal order in $N$) $${𝒪}_N\oplus {𝒪}_N\oplus \mathbb{Z}\Gamma {\cong }_{\mathbb{Z}\Gamma }\oplus {𝒪}_N^{\text{'}}\oplus {𝒪}_{N^{\text{'}}}\oplus \mathbb{Z}\Gamma .$$

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed $\alpha \in \mathbb{Q}-{\mathbb{Z}}_{\<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{\left(\alpha \right)}\left(x\right)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n+\alpha }{n-j}\right){(-x)}^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $\alpha =0,1,\pm \frac{1}{2},-1-n$.