Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the $p^2$-rank of the class group as a quantity of relevance in the $p$-class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class groups...

For p ≡ 1 (mod 4), we prove the formula (conjectured by R. Chapman) for the determinant of the (p+1)/2 × (p+1)/2 matrix $C=\left(C_{ij}\right)$ with $C_{ij}=((j-i)/p)$.

Let $d$ be a square free integer and $L_d:=\mathbb{Q}({\zeta }_8,\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, ${\mathrm{Cl}}_2\left(L_d\right)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

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 $5(mod8)$.

For any square-free positive integer $m\equiv 10(mod16)$ with $m\ge 26$, we prove that the class number of the real cyclotomic field $\mathbb{Q}({\zeta }_{4m}+{\zeta }_{4m}^{-1})$ is greater than $1$, where ${\zeta }_{4m}$ is a primitive $4m$th root of unity.

The class numbers h⁺ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime...