Related papers: Generation of ring class fields by eta-quotients

The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie…

Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Let $h_E$ be the Weber function on certain elliptic curve $E$ with complex multiplication by $\mathcal{O}_K$. We show that if $N$ ($>1$) is an integer…

We give an elementary approach to studying whether rings of $S$-integers in complex quadratic fields are Euclidean with respect to the $S$-norm.

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition…

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose…

In this paper, we determine the isomorphic classes of Morita equivalent subalgebras of irrational rotation algebras. It is based on the solution of the quadratic Diophantine equations. We determine the irrational rotation algebras that have…

We propose constructions of codes over quotient rings of Eisenstein integers equipped with the Euclidean, square Euclidean, and hexagonal distances as a generalization of codes over Eisenstein integer fields. By set partitioning, we…

We prove the triviality of the Grothendieck ring of a integer-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K^2 to itself minus a point. When we specialize to…

In connection with Eisenstein series for the principal congruence subgroup $\Gamma(n)$, Hecke introduced certain numbers, of which he said that they are rational and cumbersome to calculate. We show, however, that these numbers are…

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a…

We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we determine which hyperfields of order at most 4 are quotients…

Positively graded algebras are fairly natural objects which are arduous to be studied. In this article we query quotients of non-standard graded polynomial rings with combinatorial and commutative algebra methods.

In this article we give an algorithm for determining the generators and relations for the rings of semi-invariant functions on irreducible components of representation spaces for gentle string algebras. These rings of semi-invariants turn…

We previously showed that the inverse limit of standard-graded polynomial rings with perfect coefficient field is a polynomial ring, in an uncountable number of variables. In this paper, we show that the same result holds with arbitrary…

In these notes we investigate the rings of real polynomials in four variables, which are invariant under the action of the reflectiongroups [3,4,3] and [3,3,5]. It is well known that they are rationally generated in degree 2,6,8,12 and…

This paper introduces a novel approach to the axiomatic theory of quadratic forms. We work internally in a category of certain partially ordered sets, subject to additional conditions which amount to a strong form of local presentability.…

We construct 3-dimensional once-Extended Topological Quantum Field Theories (ETQFTs for short) out of (possibly non-semisimple) modular categories, and we explicitly identify linear categories and functors in their image. The circle…

We give an algorithm that constructs a minimal set of polynomials defining all extension of a $(\pi)$-adic field with given, inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of…

We investigate properties of zeta functions of polynomial rings and their quotients, generalizing and extending some classical results about Dedekind zeta functions of number fields. By an application of Delange's version of the Ikehara…

We remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be…