This article confirms a consequence of the non-abelian Iwasawa main conjecture. It is proved that under a technical condition the étale cohomology groups $H^1({𝒪}_K[1/S],H^i(\overline{X},{\mathbb{Q}}_p\left(j\right)))$, where $X\to Spec{𝒪}_K[1/S]$ is a smooth, projective scheme, are generated by twists of norm compatible units in a tower of number fields associated to $H^i(\overline{X},{\mathbb{Z}}_p\left(j\right))$. Using the “Bloch-Kato-conjecture” a similar result is proven for motivic cohomology with finite coefficients.

We give a necessary condition for a surjective representation Gal$(\overline{\mathbb{Q}}/\mathbb{Q})\to {\mathrm{PGL}}_2\left({𝔽}_3\right)$ to arise from the $3$-torsion of a $\mathbb{Q}$-curve. We pay a special attention to the case of quadratic $\mathbb{Q}$-curves.

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 G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition $Gal(k{₂}^{\left(2\right)}/k)\simeq G$, where $k{₂}^{\left(2\right)}$ is the second Hilbert 2-class field of k.

Let $d$ be an odd square-free integer, $m\ge 3$ any integer and $L_{m,d}:=\mathbb{Q}({\zeta }_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m,d}$ having an odd class number. Furthermore, using the cyclotomic ${\mathbb{Z}}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m,d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5(mod8)$.

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 $\alpha =0,1,\pm \frac{1}{2},-1-n$.

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 $L/\mathbb{Q}$.

For the cyclotomic ${\mathbb{Z}}_2$-extension $k_{\infty }$ of an imaginary quadratic field $k$, we consider the Galois group $G\left(k_{\infty }\right)$ of the maximal unramified pro-$2$-extension over $k_{\infty }$. In this paper, we give some families of $k$ for which $G\left(k_{\infty }\right)$ is a metabelian pro-$2$-group with the explicit presentation, and determine the case that $G\left(k_{\infty }\right)$ becomes a nonabelian metacyclic pro-$2$-group. We also calculate Iwasawa theoretically the Galois groups of $2$-class field towers of certain cyclotomic $2$-extensions.