相关论文: Monogenity in totally real extensions of imaginary…

We propose a method for generation of genuine multipartite entangled states in a short-range Ising spin chain with periodic global pulses of magnetic field. We consider an integrable and a non-integrable Floquet system that are periodic in…

This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a…

We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our…

Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques for the construction of symmetry--adapted polynomials in the symmetry…

Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a…

Let $K=\Q(\theta)$ be a number field generated by a complex root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Ga\'al studied the…

Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.

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 $\mathbb Q(\sqrt{p})$,…

We give necessary and sufficient conditions for the Kummer extension $K:=\mathbb{Q}\left(\zeta_n,\sqrt[n]{\alpha}\right)$ to be monogenic over $\mathbb{Q}(\zeta_n)$ with $\sqrt[n]{\alpha}$ as a generator, i.e., for…

For any odd prime $p,$ we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1})$ whose class numbers are both divisible by $p.$ One of our theorems settles Iizuka's conjecture for…

For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary…

We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.

Primitive polynomials over finite fields are crucial for various domains of computer science, including classical pseudo-random number generation, coding theory and post-quantum cryptography. Nevertheless, the pursuit of an efficient…

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and…

We present an adaptation of Voronoi theory for imaginary quadratic number fields of class number greater than 1. This includes a characterisation of extreme Hermitian forms which is analogous to the classic characterisation of extreme…

We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally…

We provide a simple way to add, multiply, invert, and take traces and norms of algebraic integers of a number field using integral matrices. With formulas for the integral bases of the ring of integers of at least a significant proportion…

Suppose we are given black-box access to a finite ring R, and a list of generators for an ideal I in R. We show how to find an additive basis representation for I in poly(log |R|) time. This generalizes a quantum algorithm of Arvind et al.…

Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality \[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same imaginary…

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…