Computation of Galois groups of rational polynomials

Galois theory is developed using elementary polynomial and group algebra. The method follows closely the original prescription of Galois, and has the benefit of making the theory accessible to a wide audience. The theory is illustrated by a solution in radicals of lower degree polynomials, and the standard result of the insolubility in radicals of the general quintic and above. This is augmented by the presentation of a general solution in radicals for all polynomials when su...

In this article we present a new method to obtain polynomial lower bounds for Galois orbits of torsion points of one dimensional group varieties.

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion...

In this paper we introduce a new method for finding Galois groups by computer. This is particularly effective in the case of Galois groups of p-extensions ramified at finitely many primes but unramified at the primes above p. Such Galois groups have been regarded as amongst the most mysterious objects in number theory. Very little has hitherto been discovered regarding them despite their importance in studying p-adic Galois representations unramified at p. The conjectures of ...

Rational transformations of polynomials are extensively studied in the context of finite fields, especially for the construction of irreducible polynomials. In this paper, we consider the factorization of rational transformations with (normalized) generators of the field $K(x)^G$ of $G$-invariant rational functions for $G$ a finite subgroup of $\operatorname{PGL}_2(K)$, where $K$ is an arbitrary field. Our main theorem shows that the factorization is related to a well-known g...

We give a deterministic polynomial-time algorithm to check whether the Galois group $\Gal{f}$ of an input polynomial $f(X) \in \Q[X]$ is nilpotent: the running time is polynomial in $\size{f}$. Also, we generalize the Landau-Miller solvability test to an algorithm that tests if $\Gal{f}$ is in $\Gamma_d$: this algorithm runs in time polynomial in $\size{f}$ and $n^d$ and, moreover, if $\Gal{f}\in\Gamma_d$ it computes all the prime factors of $# \Gal{f}$.

In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can find in deterministic poly(n^{\log n},\log |k|) time "either" a nontrivial factor of f(x) "or" a nontrivial automorphism of k[x]/(f(x)) of order n. This main tool leads to various ne...

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric group. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators while the other gives informati...

The inverse problem of Galois Theory was developed in the early 1800 s as an approach to understand polynomials and their roots. The inverse Galois problem states whether any finite group can be realized as a Galois group over Q (field of rational numbers). There has been considerable progress in this as yet unsolved problem. Here, we shall discuss some of the most significant results on this problem. This paper also presents a nice variety of significant methods in connectio...

Choose a polynomial $f$ uniformly at random from the set of all monic polynomials of degree $n$ with integer coefficients in the box $[-L,L]^n$. The main result of the paper asserts that if $L=L(n)$ grows to infinity, then the Galois group of $f$ is the full symmetric group, asymptotically almost surely, as $n\to \infty$. When $L$ grows rapidly to infinity, say $L>n^7$, this theorem follows from a result of Gallagher. When $L$ is bounded, the analog of the theorem is open, ...