An ABC construction of number fields

  title={An ABC construction of number fields},
  author={D. Roberts},
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 extremal solutions of the matrix equation ABC = I. The second step involves extremal polynomial solutions of the equation A(x) + B(x) + C(x) = 0. The third step involves integer solutions of the generalized Fermat equation axp + byq + czr = 0. We concentrate here on details associated to the third step and give… Expand
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 ofExpand
Nonsolvable Polynomials with Field Discriminant 5<sup>a</sup>
We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±pA for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group ofExpand
Chebyshev covers and exceptional number fields
We define rational functions Tm,n(x) and Um,n(x) in Q(x) by simple explicit formulas involving the classical Chebyshev polynomials tw(x) and uw(x). We show that these functions, viewed as covers ofExpand
Division polynomials with Galois group SU3(3).2 = G2(2)
We use a rigidity argument to prove the existence of two related degree 28 covers of the projective plane with Galois group \(SU_{3}(3).2\mathop{\cong}G_{2}(2)\). Constructing correspondingExpand
Period computations for covers of elliptic curves
This article constructs algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data, and conjecture which algebraic numbers the coefficients are, and proves this conjecture to be correct. Expand
Number Fields with Discriminant ±2 a 3 b and Galois Group A n or S n
The authors present three-point and four-point covers having bad reduction at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 12, 18, 28, and 33. By specializing these covers, theyExpand
The authors present three-point and four-point covers having bad reduction at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 12, 18, 28, and 33. By specializing these covers, theyExpand
Polynomials with prescribed bad primes
We tabulate polynomials in Z[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and infinity. WeExpand
Covers of Elliptic Curves with Unique, Totally Ramified Branch Points
A well-known and difficult problem in computational number theory and algebraic geometry is to write down equations for branched covers of algebraic curves with specified monodromy type. In thisExpand
Galois number fields with small root discriminant
Abstract We pose the problem of identifying the set K ( G , Ω ) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632 . We definitivelyExpand


Groups as Galois Groups: An Introduction
Part 1. The Basic Rigidity Criteria: 1. Hilbert's irreducibility theorem 2. Finite Galois extensions of C (x) 3. Descent of base field and the rigidity criterion 4. Covering spaces and theExpand
Sextic Number Fields with Discriminant - j 2 a 3 b
Complete lists of number fields, of given degree n and unramified outside a given finite set S of primes, are both of intrinsic interest and useful in some applications. For degrees n ≤ 5 and S = {∞,Expand
Shimura Curve Computations
  • N. Elkies
  • Mathematics, Computer Science
  • ANTS
  • 1998
Some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them are given, and a list of open questions that may point the way to further computational investigation of these curves are illustrated. Expand
Polynomials for Primitive Nonsolvable Permutation Groups of Degree d ≤ 15
  • Gunter Malle
  • Computer Science, Mathematics
  • J. Symb. Comput.
  • 1987
The determination of polynomials over @?(t) with a given primitive nonsolvable permutation group of degree d =< 15 as Galois group is completed. Sections 1-3 deal with the remaining three casesExpand
Topics in Galois Theory
This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensionsExpand
Computing division polynomials
Recurrence relations for the coefficients of the nth division polynomial for elliptic curves are presented. These provide an algorithm for computing the general division polynomial without usingExpand
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2014 A bibliography of recent papers on lower bounds for discriminants of number fields and related topics is presented. Some of the main methods, results, and open problems are discussed.
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Argument. Although the arithmetic abc conjecture is a great mystery, its algebraic counterpart is a rather easy theorem. It looks like it was rst noticed by W. W. Stothers (cf [21]). Later on it wasExpand
Rigid Local Systems
Riemann introduced the concept of a "local system" on P1-{a finite set of points} nearly 140 years ago. His idea was to study "n"th order linear differential equations by studying the rank "n" localExpand
Faltings plus epsilon , Wiles plus epsilon , and the Generalized Fermat Equation
Wiles’ proof of Fermat’s Last Theorem puts to rest one of the most famous unsolved problems in mathematics, a question that has been a wellspring for much of modern algebraic number theory. WhileExpand