Related papers: Note on some p-invariants of Q(N^{1/p}) using refl…

In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided…

In the paper "An Abelian Loop for Non-Composites" (arXiv:110.14716), we introduced a group-like structure consisting of odd prime numbers and 1, with properties that allowed us to prove analogous results to well known theorems in Number…

We use Galois cohomology to study the $p$-rank of the class group of $\mathbf{Q}(N^{1/p})$, where $N \equiv 1 \bmod{p}$ is prime. We prove a partial converse to a theorem of Calegari--Emerton, and provide a new explanation of the known…

For any extension of commutative rings $A\subseteq B$, by using invertible ideals, we first define an Abelian group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the main properties of this group. Among…

Let $\mathfrak{n} = \mathfrak{p}^r$ be a prime power ideal of $\mathbb{F}_q[T]$ with $r \geq 2$. We study the rational torsion subgroup $\mathcal{T}(\mathfrak{p}^r)$ of the Drinfeld modular Jacobian $J_0(\mathfrak{p}^r)$. We prove that the…

Let $p\geq 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ form an algebraically closed subfield $\mathbb{L}_p^{\operatorname{ha}}$. In this article, we clarify the relations among the fields…

A class of $C^*$-algebras, to be called those of generalized tracial rank one, is introduced, and classified by the Elliott invariant. A second class of unital simple separable amenable $C^*$-algebras, those whose tensor products with…

Let p be a prime number. It is not known if every finite p-group of rank n>1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the family of finite p-groups which contains all…

Suppose that $G$ is a finite group such that $\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial…

In this paper, we prove some results on the asymptotic behavior arising in modular representation theory over abelian $p$-groups. First, we embed the representation ring of a cyclic $p$-group into a real algebra of functions. Second, we…

Let $K_{m}=\Bbb{Q}(\zeta_{m})$ where $\zeta_{m}$ is a primitive $m$th root of unity. Let $p>2$ be prime and let $C_{p}$ denote the group of order $p.$ The ring of algebraic integers of $K_{m}$ is $\Cal{O}_{m}=\Bbb{Z}[\zeta_{m}].$ Let…

I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$.

Let $p$ be an odd prime and let $G$ be a non-abelian finite $p$-group of exponent $p^2$ with three distinct characteristic subgroups, namely $1$, $G^p$, and $G$. The quotient group $G/G^p$ gives rise to an anti-commutative ${\mathbb…

This paper is devoted to the classification and studying properties of complex unital $3$-dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type $(2, 1)$ and two algebras…

If p is a prime, then the numbers 1, 2, ..., p-1 form a group under multiplication modulo p. A number g that generates this group is called a primitive root of p; i.e., g is such that every number between 1 and p-1 can be written as a power…

We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime number $p$. For the proof, we use the fundamental unit of $\mathbb Q(\sqrt{p})$,…

Let p>3 be a prime number and let r be an integer with 1<r<p-1. For each r, let moreover G_r denote the unique quotient of the maximal class pro-p group of size p^{r+1}. We show that the mod-p cohomology ring of G_r has depth one and that,…

We prove a conjecture of J. Carlson, N. Mazza and J. Th\'evenaz; namely, we will prove that if $G$ is a finite $p$-nilpotent group which contains a non-cyclic elementary Abelian $p$-subgroup and $k$ is an algebraically closed field of…

For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density is…

Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev…