One of the important problems in algebraic number theory is to study the monogenity of number fields. Monogenic number fields arise from the roots of monogenic polynomials. In this article, we deal with the problem of monogenity of the composition of two monic polynomials having integer coefficients. We provide necessary and sufficient conditions for the composition to be monogenic together with a further sufficient condition. At the end of the paper, we construct an infinite tower of monogenic number fields.

Let G be a finite nilpotent group and $n\in \{3,4, 5\}$. Consider $S_n\times G$ as a subgroup of $S_n\times S_{|G|}\subset S_{n|G|}$, where G embeds into the second factor of $S_n\times S_{|G|}$ via the regular representation. Over any number field k, we prove the strong form of Malle’s conjecture (cf. Malle (2002, Journal of Number Theory 92, 315–329)) for $S_n\times G$ viewed as a subgroup of $S_{n|G|}$. Our result requires that G satisfies some mild conditions.

We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions $K/\mathbb{Q}$ with Galois group isomorphic to $A_4$, $S_4$, $A_5$, and dihedral groups of order $2p^n$ for certain prime powers $p^n$. In particular, when $K/\mathbb{Q}$ is a Galois extension with Galois group $G$ isomorphic to $A_4$, $S_4$ or $A_5$, we give necessary and sufficient conditions for the ring of integers $\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra $\mathbb{Q}[G]$.

We apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^*$-algebras. First, we give a new proof of non-inner conjugacy of the diagonal MASA ${\mathcal {D}}_E$ to its non-trivial image under a quasi-free automorphism, where $E$ is a finite transitive graph. Changing graphs representing the algebras, this result applies to some non quasi-free automorphisms as well. Then, we exhibit a large class of MASAs in the Cuntz algebra ${\mathcal {O}}_n$ that are not inner conjugate to the diagonal ${\mathcal {D}}_n$.

Let $K={\mathbf {Q}}(\theta )$ be an algebraic number field with $\theta$ a root of an irreducible polynomial $x^5+ax+b\in {\mathbf {Z}}[x]$. In this paper, for every rational prime $p$, we provide necessary and sufficient conditions on $a,\,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,\,~b$ for which $K$ is non-monogenic. We illustrate our results through examples.

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

The notion of $\theta $-congruent numbers is a generalisation of congruent numbers where one considers triangles with an angle $\theta $ such that $\cos \theta $ is a rational number. In this paper we discuss a criterion for a natural number to be $\theta $-congruent over certain real number fields.

In this paper we describe how to compute smallest monic polynomials that define a given number field  $\mathbb{K}$ . We make use of the one-to-one correspondence between monic defining polynomials of  $\mathbb{K}$ and algebraic integers that generate  $\mathbb{K}$ . Thus, a smallest polynomial corresponds to a vector in the lattice of integers of  $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of  $\mathbb{K}$ . This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

In previous work, Ohno conjectured, and Nakagawa proved, relations between thecounting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

We describe an online database of number fields which accompanies this paper. The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.

For each solvable Galois group which appears in degree $9$ and each allowable signature, we find polynomials which define the fields of minimum absolute discriminant.