What should be assumed about the integral polynomials $f₁\left(x\right),...,f_k\left(x\right)$ in order that the solvability of the congruence $f₁\left(x\right)f₂\left(x\right)\cdots f_k\left(x\right)\equiv 0\left(modp\right)$ for sufficiently large primes p implies the solvability of the equation $f₁\left(x\right)f₂\left(x\right)\cdots f_k\left(x\right)=0$ in integers x? We provide some explicit characterizations for the cases when $f_j\left(x\right)$ are binomials or have cyclic splitting fields.

The aim of this paper is to prove the following Theorem Theorem Let $K$ be an octic subfield of the field $Q({\zeta }_p+{\zeta }_p^{-1})$ and let $p=n^4+16$ be prime. Then $p$ divides $h_K$ if and only if $p$ divides $B_j$ for some $j=\frac{p-1}{8}$, $3\frac{p-1}{8}$, $5\frac{p-1}{8}$, $7\frac{p-1}{8}$.

Galois extensions with various metacyclic Galois groups are constructed by means of a Kummer theory arising from an isogeny of certain algebraic tori. In particular, our method enables us to construct algebraic tori parameterizing metacyclic extensions.

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.

We study Tate’s refinement for a conjecture of Gross on the values of abelian $L$-function at $s=0$ and formulate its generalization to arbitrary cyclic extensions. We prove that our generalized conjecture is true in the case of number fields. This in particular implies that Tate’s refinement is true for any number field.

Let $p$ be an odd prime, $\chi$ an odd, $p$-adic Dirichlet character and $K$ the cyclic imaginary extension of $\mathbf{Q}$ associated to $\chi$. We define a “$\chi$-part” of the Sylow $p$-subgroup of the class group of $K$ and prove a result relating its $p$-divisibility to that of the generalized Bernoulli number $B_{1,{\chi }^{-1}}$. This uses the results of Mazur and Wiles in Iwasawa theory over $\mathbf{Q}$. The more difficult case, in which $p$ divides the order of $\chi$ is our chief concern. In this case the result is new and confirms an earlier conjecture of G....

General methods from [3] are applied to give good upper bounds on the Euclidean minimum of real quadratic fields and totally real cyclotomic fields of prime power discriminant.