Topic
Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
Papers published on a yearly basis
Papers
More filters
Posted Content•
TL;DR: In this article, a formalism for counting integer and rational solutions to polynomial equations with rational coefficients is given, where the problem of counting is described by two elliptic curves and a map between them.
Abstract: A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational solutions to $y^2=P(x)$ is facilitated by another elliptic curve with integral coefficients. The problem of counting is described by two elliptic curves and a map between them.
6 citations
TL;DR: In this paper, the authors show that the ring of integers in a finite extension of the rationals is a half-factorial domain (HFD) if and only if the class number of R is less than or equal to 2.
6 citations
TL;DR: In this paper, it was shown that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero.
Abstract: In this paper we prove undecidability of finite systems of equations in free Lie algebras of rank at least three over an arbitrary field. We show that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero.
6 citations
Posted Content•
TL;DR: In this paper, the authors generalize the definition and properties of root systems to complex reflection groups, where roots become rank one projective modules over the ring of integers of a number field k.
Abstract: We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation. In the case of spetsial reflection groups, we generalize as well the definition and properties of bad primes.
6 citations
TL;DR: In this paper, the Montes algorithm was used to show that a root of polynomials (i.e., a polynomial in the families f(a,b) and g(c,d) can be used as a generator for a power integral basis of the ring of integers.
Abstract: Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\mid a$, and the integers $b$, $d$, $256d-27c^4$, and $\dfrac{256b^3-27a^4}{\gcd(256b^3,27a^4)}$ are all square-free. Using the Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a monogenic extension of $\mathbb{Q}$ and serves as a generator for a power integral basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generating monogenic $S_4$ fields within the families $f_{b,b}(x)$ and $g_{1,d}(x)$.
6 citations