Computation of Galois groups of rational polynomials

Fieker, Claus; Klüners, Jürgen

doi:10.1112/s1461157013000302

Cited by 24 publications

(38 citation statements)

References 20 publications

(35 reference statements)

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“…Galois groups of irreducible polynomials over Q(t). The article [7] describes an algorithm to compute Galois groups of irreducible polynomials over Q. As noted in [7] Section 7.7, this algorithm can be adjusted to compute Galois groups of polynomials over fields other than the rational field.…”

Section: Computation Of Galois Groups Over Q(t)mentioning

confidence: 99%

“…The article [7] describes an algorithm to compute Galois groups of irreducible polynomials over Q. As noted in [7] Section 7.7, this algorithm can be adjusted to compute Galois groups of polynomials over fields other than the rational field. For example, [23,22] discusses this for polynomials over global rational and algebraic function fields.…”

Section: Computation Of Galois Groups Over Q(t)mentioning

confidence: 99%

“…To achieve step (1), one must begin by computing a permutation representation of the Galois group G; this can be done using methods of Fieker and Klüners [7]. Though these authors mainly discuss the case of irreducible polynomials over Q, their methods can be extended to work more generally.…”

Section: Introductionmentioning

confidence: 99%

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Galois groups over rational function fields and Explicit Hilbert Irreducibility

Sutherland

2021

Journal of Symbolic Computation

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x] be a polynomial in two variables with rational coefficients, and let G be the Galois group of P over the field Q(t). It follows from Hilbert's Irreducibility Theorem that for most rational numbers c the specialized polynomial P (c, x) has Galois group isomorphic to G and factors in the same way as P . In this paper we discuss methods for computing the group G and obtaining an explicit description of the exceptional numbers c, i.e., those for which P (c, x) has Galois group different from G or factors differently from P . To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics.

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Section: Computation Of Galois Groups Over Q(t)mentioning

confidence: 99%

Section: Computation Of Galois Groups Over Q(t)mentioning

confidence: 99%

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Galois groups over rational function fields and Explicit Hilbert Irreducibility

Sutherland

2021

Journal of Symbolic Computation

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“…Recall that the height of a rational number a/b with gcd(a, b) = 1 is given by max(|a|, |b|). Fixing a height bound h, it a straightforward procedure to construct the set B(h) of all rational numbers having height at most h. One can then construct all the polynomials Φ n (c, x) for c ∈ B(h), compute their Galois groups G n,c (for instance, using the algorithm of Fieker and Klüners [9], which is implemented in Magma), and check whether G n,c ∼ = (Z/nZ) S r . The cost of carrying out this computation grows quickly with n, given the large degree of Φ n .…”

Section: The Exceptional Sets E Nmentioning

confidence: 99%

A finiteness theorem for specializations of dynatomic polynomials

Krumm

2019

Alg. Number Th.

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Let t and x be indeterminates, let φ(x) = x 2 + t ∈ Q(t) [x], and for every positive integer n let Φn(t, x) denote the n th dynatomic polynomial of φ. Let Gn be the Galois group of Φn over the function field Q(t), and for c ∈ Q let Gn,c be the Galois group of the specialized polynomial Φn(c, x). It follows from Hilbert's irreducibility theorem that for fixed n we have Gn ∼ = Gn,c for every c outside a thin set En ⊂ Q. By earlier work of Morton (for n = 3) and the present author (for n = 4), it is known that En is infinite if n ≤ 4. In contrast, we show here that En is finite if n ∈ {5, 6, 7, 9}. As an application of this result we show that, for these values of n, the following holds with at most finitely many exceptions: for every c ∈ Q, more than 81% of prime numbers p have the property that the polynomial x 2 + c does not have a point of period n in the p-adic field Qp.

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“…Since Stauduhar developed an interesting practical algorithm [7] for the computation of Galois groups, there have been a number of other algorithms described but these have mostly been specific to irreducible polynomials over the rational field. We consider the recent algorithm of Fieker and Klüners [4] and describe how to adjust this algorithm so that it can be used to compute Galois groups of polynomials over characteristic p function fields, including when the characteristic is two (in which case replacement invariants were required). Further, we provide an algorithm to compute Galois groups of reducible polynomials, including those over function fields of characteristic p.…”

mentioning

confidence: 99%

Algorithms for Galois Extensions of Global function Fields

Sutherland

2016

Bull. Aust. Math. Soc.

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This thesis considers some computational problems in cyclic Galois extensions of global function fields. We investigate the efficient computation of integral closures, or maximal orders, in cyclic extensions of global fields and the determination of Galois groups for polynomials over global function fields.Global function fields, which are finite separable extensions of a global rational function field, are interesting because they provide a basis for designing efficient algorithms for algebraic curves. Applications of curves and function fields arise in coding theory and cryptography. Efficient computation with curves or function fields is necessary for efficient construction of codes.In this thesis, we consider function fields from the number theory point of view, and take advantage of algorithms for number fields (such as those in [2]), which can be used analogously for function fields. Algorithms are stated generally, so it is irrelevant whether the field is a number or a function field and whether or not it is represented as an extension of another algebraic (function) field.Some tasks have a rich history for number fields but have only recently sparked interest for function fields. These include the development of methods to efficiently compute integral closures (an analogue of Z), Galois groups, class groups and unit groups, which are the four most important tasks of number theory considered by Zassenhaus [6]. We investigate two of these. Integral closures can be used to compute class groups, unit groups and Galois groups, which are the other three important tasks.Also interesting for function fields is the computation of Riemann-Roch spaces of divisors and the genus. The calculation of a basis for a Riemann-Roch space can use the representation of a divisor by ideals of integral closures. Improving the efficiency

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Computation of Galois groups of rational polynomials

Cited by 24 publications

References 20 publications

Galois groups over rational function fields and Explicit Hilbert Irreducibility

Galois groups over rational function fields and Explicit Hilbert Irreducibility

A finiteness theorem for specializations of dynatomic polynomials

Algorithms for Galois Extensions of Global function Fields

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