Let $K$ be a field of degree $n$ over ${\mathbf{Q}}_p$, the field of rational $p$-adic numbers, say with residue degree $f$, ramification index $e$ and differential exponent $d$. Let $O$ be the ring of integers of $K$ and $\mathit{P}$ its unique prime ideal. The trace and norm maps for $K/{\mathbf{Q}}_p$ are denoted $Tr$ and $N$, respectively. Fix $q=p^r$, a power of a prime $p$, and let $\eta$ be a numerical character defined modulo $q$ and of order $o\left(\eta \right)$. The character $\eta$ extends to the ring of $p$-adic integers ${\mathbb{Z}}_p$ in the natural way; namely $\eta \left(u\right)=\eta \left(\tilde{u}\right)$, where $\tilde{u}$ denotes the residue...

Let $p\ne 2$ be a prime and let $G$ be a $p$-group of matrices in ${\mathrm{SL}}_n\left(\mathbb{Z}\right)$, for some integer $n$. In this paper we show that, when $n\<3(p-1)$, a certain subgroup of the cohomology group $H^1(G,{𝔽}_p^n)$ is trivial. We also show that this statement can be false when $n\ge 3(p-1)$. Together with a result of Dvornicich and Zannier (see []), we obtain that any algebraic torus of dimension $n\<3(p-1)$ enjoys a local-global principle on divisibility by $p$.

We provide a generalization of Scholz’s reciprocity law using the subfields $K_{2^{t-1}}$ and $K_{2^t}$ of $\mathbb{Q}\left({\zeta }_p\right)$, of degrees $2^{t-1}$ and $2^t$ over $\mathbb{Q}$, respectively. The proof requires a particular choice of primitive element for $K_{2^t}$ over $K_{2^{t-1}}$ and is based upon the splitting of the cyclotomic polynomial ${\Phi }_p\left(x\right)$ over the subfields.

A groupoid is alternative if it satisfies the alternative laws $x\left(xy\right)=\left(xx\right)y$ and $x\left(yy\right)=\left(xy\right)y$. These laws induce four partial maps on ${\mathbb{N}}^{+}\times {\mathbb{N}}^{+}$ $$(r,s)\mapsto (2r,s-r),(r-s,2s),(r/2,s+r/2),(r+s/2,s/2),$$ that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that $n$th powers in a free alternative groupoid on one generator are well-defined if and only if $n\le 5$. We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses. ...

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)....