Galois theory of Salem polynomials

2009 ◽  
Vol 148 (1) ◽  
pp. 47-54 ◽  
Author(s):  
CHRISTOS CHRISTOPOULOS ◽  
JAMES MCKEE
Keyword(s):  
Unit Circle ◽  
Galois Group ◽  
Galois Theory ◽  
Minimal Polynomial ◽  
Galois Groups ◽  
Cyclotomic Polynomials ◽  
Salem Number ◽  
Extra Structure

AbstractLet f(x) ∈ [x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1/r1, r2, 1/r2, . . ., rd, 1/rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 + 1/r1, . . ., rd + 1/rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg ≅ Gf/N, where N is isomorphic to a subgroup of C2d. In a naive sense, the generic case is Gf ≅ C2d ⋊ Sd, with N ≅ C2d and Gg ≅ Sd. When f(x) has extra structure this may be reflected in the Galois group, and it is not always true even that Gf ≅ N ⋊ Gg. For example, for cyclotomic polynomials f(x) = Φn(x) it is known that Gf ≅ N ⋊ Gg if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4.In this paper we deal with irreducible reciprocal monic polynomials f(x) ∈ [x] that are ‘close’ to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f(x) has degree at least 4, this means that f(x) is the minimal polynomial of a Salem number. We show that in this case one always has Gf ≅ N ⋊ Gg, and moreover that N ≅ C2d or C2d−1, with the latter only possible if d is odd.

scholarly journals Computation of Galois groups of rational polynomials

2014 ◽  
Vol 17 (1) ◽  
pp. 141-158 ◽  
Author(s):  
Claus Fieker ◽  
Jürgen Klüners
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Large Degree ◽  
Rational Numbers ◽  
Galois Groups ◽  
Multivariate Polynomial ◽  
Theoretical Methods ◽  
Algorithmic Solution ◽  
Rational Polynomials ◽  
General Method

AbstractComputational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups $H<G$, a $G$-relative $H$-invariant, that is a multivariate polynomial $F$ that is $H$-invariant, but not $G$-invariant. While generic, theoretical methods are known to find such $F$, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.

scholarly journals Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
A. Esterov ◽  
L. Lang
Keyword(s):  
Galois Group ◽  
Wreath Product ◽  
Galois Theory ◽  
General System ◽  
Wreath Products ◽  
Galois Groups ◽  
Complex Polynomial ◽  
Polynomial Equations ◽  
Sparse Polynomial ◽  
System Of Polynomial Equations

AbstractWe introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial $$f(x) = c_0 + c_1 x^{d_1} + \cdots + c_k x^{d_k}$$ f ( x ) = c 0 + c 1 x d 1 + ⋯ + c k x d k by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable $$y=x^d$$ y = x d , and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over $$d_k/d$$ d k / d elements and $${\mathbb {Z}}/d{\mathbb {Z}}$$ Z / d Z . We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.

scholarly journals A classification of all 1-Salem graphs

2014 ◽  
Vol 17 (1) ◽  
pp. 582-594 ◽  
Author(s):  
Lee Gumbrell ◽  
James McKee
Keyword(s):  
New York ◽  
Minimal Polynomial ◽  
Cyclotomic Polynomials ◽  
Critical Interval ◽  
Combinatorial Structures ◽  
Induced Subgraph ◽  
Combinatorial Objects ◽  
Salem Number ◽  
Bipartite Case

AbstractOne way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph $G$ has an associated reciprocal polynomial $R_{G}$, and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is $R_{G}$ a product of cyclotomic polynomials (giving the cyclotomic graphs)? (b) when does $R_{G}$ have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trivial Salem graphs)? Cyclotomic graphs were classified by Smith (Combinatorial structures and their applications, Proceedings of Calgary International Conference, Calgary, AB, 1969 (eds R. Guy, H. Hanani, H. Saver and J. Schönheim; Gordon and Breach, New York, 1970) 403–406); the maximal connected ones are known as Smith graphs. Salem graphs are ‘spectrally close’ to being cyclotomic, in that nearly all their eigenvalues are in the critical interval $[-2,2]$. On the other hand, Salem graphs do not need to be ‘combinatorially close’ to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny.We define an $m$-Salem graph to be a connected Salem graph $G$ for which $m$ is minimal such that there exists an induced cyclotomic subgraph of $G$ that has $m$ fewer vertices than $G$. The $1$-Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a $1$-Salem graph as an induced subgraph, so these $1$-Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all $1$-Salem graphs: in the non-bipartite case there are $25$ infinite families and $383$ sporadic examples.

Effective Galois theory

1981 ◽  
Vol 46 (2) ◽  
pp. 385-392 ◽  
Author(s):  
Peter La Roche
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Galois Groups ◽  
Profinite Group ◽  
Classical Literature ◽  
Profinite Groups ◽  
Algebraic Numbers ◽  
Recursively Enumerable ◽  
Free Profinite Group ◽  
Infinite Degree

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups (i.e. totally disconnected, compact, Hausdorff groups). Metakides and Nerode [7] produced two recursively presented algebraic extensionsK⊂Fof the rationals such thatFis abelian,Fis of infinite degree overK, and the Galois group ofFoverK, although of cardinalityc, has only one recursive element (viz. the identity). This indicated the limits of effectiveness for Krull's theory. (The Galois theory offiniteextensions is completely effective.) Nerode suggested developing a natural effective version of Krull's theory (done here in §1).It is evident from the classical literature that the free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield (done here in §2). The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield.

scholarly journals A note on Galois groups of algebraic closures

1976 ◽  
Vol 21 (1) ◽  
pp. 12-15 ◽  
Author(s):  
Don Krakowski
Keyword(s):  
Galois Group ◽  
Fundamental Theorem ◽  
Inverse Limit ◽  
Galois Theory ◽  
Algebraic Closure ◽  
Galois Groups ◽  
Point Of View ◽  
Roots Of Unity ◽  
Profinite Groups ◽  
Profinite Topology

A group will be called full if it is the Galois group of an algebraic closure of a field. In this paper we first investigate full Abelian groups and classify them. Then we examine full groups from the point of view of how we can operate on them and still maintain the property of being full. Of course, by the fundamental theorem of the infinite Galois theory, closed subgroups (with standard profinite topology) of full groups are full. In general, products of full groups are notfull (for example, Z2 × Z2 is not full, by a theorem of Artin and Schreier (1927)); however we produce a set of groups which can always be attached as direct factors to full groups and still retain full groups. For the definition and basic properties of profinite groups we refer the reader to Cassels and Frohlich (1967). It has been shown by Leptin (1955) and independently by the author that profinite groups are Galois groups. The author (1971) has shown that if G is profinite, then G = Gdl(k/L) where L may have any desired characteristic and contains all primitive nth roots of unity, for all n. We will denote by p the pro-p-group which is the inverse limit of cyclic p-groups. (This is the group of p-adic integers).

A galois theory for the banach algebra of continuous symmetric functions on absolute galois groups

2004 ◽  
Vol 45 (3-4) ◽  
pp. 349-358 ◽  
Author(s):  
Angel Popescu ◽  
Nicolae Popescu ◽  
Alexandru Zaharescu
Keyword(s):  
Banach Algebra ◽  
Symmetric Functions ◽  
Galois Theory ◽  
Galois Groups

scholarly journals On the Galois group of three classes of trinomials

2021 ◽  
Vol 7 (1) ◽  
pp. 212-224
Author(s):  
Lingfeng Ao ◽  
◽  
Shuanglin Fei ◽  
Shaofang Hong
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Prime Number ◽  
Galois Groups

<abstract><p>Let $ n\ge 8 $ be an integer and let $ p $ be a prime number satisfying $ \frac{n}{2} &lt; p &lt; n-2 $. In this paper, we prove that the Galois groups of the trinomials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ T_{n, p, k}(x): = x^n+n^kp^{(n-1-p)k}x^p+n^kp^{nk}, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ S_{n, p}(x): = x^n+p^{n(n-1-p)}n^px^p+n^pp^{n^2} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ E_{n, p}(x): = x^n+pnx^{n-p}+pn^2 $\end{document} </tex-math></disp-formula></p> <p>are the full symmetric group $ S_n $ under several conditions. This extends the Cohen-Movahhedi-Salinier theorem on the irreducible trinomials $ f(x) = x^n+ax^s+b $ with integral coefficients.</p></abstract>

scholarly journals Calculating Galois groups of third-order linear differential equations with parameters

2018 ◽  
Vol 20 (04) ◽  
pp. 1750038
Author(s):  
Andrei Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Galois Group ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Unipotent Radical ◽  
Third Order ◽  
Differential Galois Group ◽  
Bessel Differential Equation ◽  
Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras

2012 ◽  
Vol 55 (1) ◽  
pp. 38-47
Author(s):  
William Butske
Keyword(s):  
Galois Group ◽  
Galois Groups ◽  
Two Dimensional ◽  
Finite Algebras ◽  
Genus 2

AbstractZarhin proves that if C is the curve y2 = f (x) where Galℚ(f(x)) = Sn or An, then . In seeking to examine his result in the genus g = 2 case supposing other Galois groups, we calculate for a genus 2 curve where f (x) is irreducible. In particular, we show that unless the Galois group is S5 or A5, the Galois group does not determine .

scholarly journals On Galois groups of class two extensions over the rational number field

1979 ◽  
Vol 75 ◽  
pp. 121-131 ◽  
Author(s):  
Susumu Shirai
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Number Field ◽  
Direct Product ◽  
Rational Number ◽  
Nilpotency Class ◽  
Abelian Extension ◽  
Galois Groups ◽  
Classification Theory ◽  
Ray Class Field

Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l, and let f(K) be the conductor of K/Q; if l = 2, let K be complex, and if in addition f(K) ≡ 0 (mod 2), let f(K) ≡ 0 (mod 16). Denote by (K) the Geschlechtermodul of K over Q and by K̂ the maximal central l-extension of K/Q contained in the ray class field mod (K) of K. A. Fröhlich [1, Theorem 4] completely determined the Galois group of K̂ over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f(K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l. Hence we know the set of fields of nilpotency class two over Q, because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

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