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

Symmetric functions, $m$-sets, and Galois groups

1994 ◽  
Vol 63 (208) ◽  
pp. 749-749
Author(s):  
David Casperson ◽  
John McKay
Keyword(s):  
Symmetric Functions ◽  
Galois Groups

scholarly journals DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

2015 ◽  
Vol 16 (1) ◽  
pp. 59-119 ◽  
Author(s):  
Lucia Di Vizio ◽  
Charlotte Hardouin ◽  
Michael Wibmer
Keyword(s):  
Differential Equations ◽  
Characteristic Zero ◽  
Galois Theory ◽  
Structure Theorem ◽  
Special Functions ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Linear Differential

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

scholarly journals GEOMETRIC GALOIS THEORY, NONLINEAR NUMBER FIELDS AND A GALOIS GROUP INTERPRETATION OF THE IDELE CLASS GROUP

2005 ◽  
Vol 16 (06) ◽  
pp. 567-593
Author(s):  
T. M. GENDRON ◽  
A. VERJOVSKY
Keyword(s):  
Galois Theory ◽  
Class Group ◽  
Number Fields ◽  
Field Extension ◽  
Abelian Extension ◽  
Galois Groups ◽  
Algebraic Number Fields ◽  
Infinite Degree ◽  
Groups Of Automorphisms ◽  
Holomorphic Extensions

This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over ℚ, a hyperbolized adele class group [Formula: see text] is assigned to every number field K/ℚ. The projectivization of the Hardy space ℙ𝖧•[K] of graded-holomorphic functions on [Formula: see text] possesses two operations ⊕ and ⊗ giving it the structure of a nonlinear field extension of K. We show that the Galois theory of these nonlinear number fields coincides with their discrete counterparts in that 𝖦𝖺𝗅(ℙ𝖧•[K]/K) = 1 and 𝖦𝖺𝗅(ℙ𝖧•[L]/ℙ𝖧•[K]) ≅ 𝖦𝖺𝗅(L/K) if L/K is Galois. If K ab denotes the maximal abelian extension of K and 𝖢K is the idele class group, it is shown that there are embeddings of 𝖢K into 𝖦𝖺𝗅⊕(ℙ𝖧•[K ab ]/K) and 𝖦𝖺𝗅⊗(ℙ𝖧•[K ab ]/K), the "Galois groups" of automorphisms preserving ⊕ (respectively, ⊗) only.

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.

Semi-topological Galois Theory

2015 ◽  
Vol 22 (04) ◽  
pp. 687-706
Author(s):  
Hsuan-Yi Liao ◽  
Jyh-Haur Teh
Keyword(s):  
Galois Theory ◽  
Galois Groups ◽  
Braid Groups ◽  
Covering Spaces ◽  
Cyclic Groups ◽  
Inverse Galois Problem ◽  
Field Extensions ◽  
Galois Correspondence

We introduce splitting coverings to enhance the well known analogy between field extensions and covering spaces. Semi-topological Galois groups are defined for Weierstrass polynomials and a Galois correspondence is proved. Combining results from braid groups, we solve the topological inverse Galois problem. As an application, symmetric and cyclic groups are realized over ℚ.

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.

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.

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