# An ABC construction of number fields

@inproceedings{Roberts2004AnAC, title={An ABC construction of number fields}, author={D. Roberts}, year={2004} }

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

#### 15 Citations

NONSOLVABLE POLYNOMIALS WITH FIELD DISCRIMINANT 5 A

- Mathematics
- 2011

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… Expand

Nonsolvable Polynomials with Field Discriminant 5<sup>a</sup>

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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 of… Expand

Chebyshev covers and exceptional number fields

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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 of… Expand

Division polynomials with Galois group SU3(3).2 = G2(2)

- Mathematics
- 2015

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 corresponding… Expand

Period computations for covers of elliptic curves

- Computer Science, Mathematics
- Math. Comput.
- 2014

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

- Mathematics
- 2005

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, they… Expand

NUMBER FIELDSWITH DISCRIMINANT ± 2 a 3 b AND GALOIS GROUP An OR

- 2005

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, they… Expand

Polynomials with prescribed bad primes

- Mathematics
- 2014

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. We… Expand

Covers of Elliptic Curves with Unique, Totally Ramified Branch Points

- Mathematics
- 2012

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 this… Expand

Galois number fields with small root discriminant

- Mathematics
- 2007

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 definitively… Expand

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