NONSOLVABLE POLYNOMIALS WITH FIELD DISCRIMINANT 5 A

@article{Roberts2011NONSOLVABLEPW,
  title={NONSOLVABLE POLYNOMIALS WITH FIELD DISCRIMINANT 5 A},
  author={D. Roberts},
  journal={International Journal of Number Theory},
  year={2011},
  volume={7},
  pages={289-322}
}
  • D. Roberts
  • Published 2011
  • Mathematics
  • International Journal of Number Theory
We present the first explicitly known polynomials in Z(x) with nonsolvable Galois group and field discriminant of the form ±p A for p 7 a prime. Our main polyno- mial has degree 25, Galois group of the form PSL2(5) 5 .10, and field discriminant 5 69 . A closely related polynomial has degree 120, Galois group of the form SL2(5) 5 .20, and field discriminant 5 311 . We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125·5 1/12500… Expand

Tables from this paper

On the number of ramified primes in specializations of function fields over $\mathbb{Q}$
We study the number of ramified prime numbers in finite Galois extensions of $\mathbb{Q}$ obtained by specializing a finite Galois extension of $\mathbb{Q}(T)$. Our main result is a central limitExpand
On the number of ramified primes in specializations of function fields over Q
We study the number of ramified prime numbers in finite Galois extensions of Q obtained by specializing a finite Galois extension of Q(T ). Our main result is a central limit theorem for this number.Expand
Nonsolvable number fields ramified only at 3 and 5
Abstract For p=3 and p=5, we exhibit a finite nonsolvable extension of ℚ which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computationsExpand
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomialExpand
Explicit Methods for Hilbert Modular Forms
The study of modular forms remains a dominant theme in modern number theory, a consequence of their intrinsic appeal as well as their applications to a wide variety of mathematical problems. ThisExpand
ON COMPUTING BELYI MAPS par
Nous donnons un aperçu des méthodes actuelles pour le calcul des revêtements de la droite projective ramifiés sur au plus trois points, aussi appelés les morphismes de Bely̆ı. Ces méthodesExpand

References

SHOWING 1-10 OF 37 REFERENCES
An ABC construction of number fields
We describe a general three step method for constructing number fields with Lie-type Galois groups and discriminants factoring into powers of specified primes. The first step involves extremalExpand
A non-solvable Galois extension of ramified at 2 only
Abstract In this Note, we show the existence of a non-solvable Galois extension of Q which is unramified outside 2. The extension K we construct has degree 2 251 731 094 732 800 = 2 19 ( 3 ⋅ 5 ⋅ 17 ⋅Expand
Heuristics for class numbers of prime-power real cyclotomic fields
Abstract. Let h(`) denote the class number of the maximal totally real subfield Q(cos(2π/`n)) of the field of `n-th roots of unity. The goal of this paper is to show that (speculative extensions of)Expand
Hecke Algebras and Automorphic Forms
The goal of this paper is to carry out some explicit calculations of the actions of Hecke operators on spaces of algebraic modular forms on certain simple groups. In order to do this, we give theExpand
Mazur's Principle for Totally Real Fields of Odd Degree
In this paper, we prove an analogue of the result known as Mazur's Principle concerning optimal levels of mod ℓ Galois representations. The paper is divided into two parts. We begin with the studyExpand
On the computation of Galois representations associated to level one modular forms
In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l<=23 and the modular forms considered will be cusp forms of level 1Expand
On the Equations Z
We investigate integer solutions of the superelliptic equation (1) z = F (x, y), where F is a homogenous polynomial with integer coefficients, and of the generalized Fermat equation (2) Ax + By = Cz,Expand
A targeted Martinet search
TLDR
This paper considers the problem of computing all imprimitive number fields of a given degree which are unramified outside of agiven finite set of primes S by combining the techniques of targeted Hunter searches with Martinet's relative version of Hunter's theorem. Expand
Number Fields Ramified at One Prime
TLDR
The existence of G-p fields for fixed G and varying p is Westudy's conjecture. Expand
Introduction to Cyclotomic Fields
1 Fermat's Last Theorem.- 2 Basic Results.- 3 Dirichlet Characters.- 4 Dirichlet L-series and Class Number Formulas.- 5 p-adic L-functions and Bernoulli Numbers.- 5.1. p-adic functions.- 5.2. p-adicExpand
...
1
2
3
4
...