Galois groups of some iterated polynomials over cyclotomic extensions
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DOI: 10.1007/s00013-017-1120-6
Abstract: Let $$\varphi _p(z)=(z-1)^p+2-\zeta _p$$φp(z)=(z-1)p+2-ζp, where $$\zeta _p\in \bar{\mathbb {Q}}$$ζp∈Q¯ is a primitive pth root of unity. Building on previous work, we show that the nth iterate $$\varphi _p^n(z)$$φpn(z) has Galois group $$[C_p]^n$$[Cp]n, an iterated wreath… read more here.
Keywords: polynomials cyclotomic; galois groups; cyclotomic extensions; iterated polynomials ... See more keywords
Numerical Computation of Galois Groups
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DOI: 10.1007/s10208-017-9356-x
Abstract: The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. We give numerical methods to compute the Galois group and study it when… read more here.
Keywords: numerical computation; galois; group; galois groups ... See more keywords
Galois groups for integrable and projectively integrable linear difference equations
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DOI: 10.1016/j.jalgebra.2017.02.032
Abstract: We consider first-order linear difference systems over C(x)C(x), with respect to a difference operator σ that is either a shift σ:x↦x+1σ:x↦x+1, q -dilation σ:x↦qxσ:x↦qx with q∈C×q∈C× not a root of unity, or Mahler operator σ:x↦xqσ:x↦xq… read more here.
Keywords: groups integrable; difference; integrable projectively; galois groups ... See more keywords
ON GALOIS GROUPS OF POWER COMPOSITIONAL NONIC POLYNOMIALS
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DOI: 10.1017/s0004972725000048
Abstract: Abstract Let $g(x)=x^3+ax^2+bx+c$ and $f(x)=g(x^3)$ be irreducible polynomials with rational coefficients, and let $ {\mathrm{Gal}}(f)$ be the Galois group of $f(x)$ over $\mathbb {Q}$ . We show $ {\mathrm{Gal}}(f)$ is one of 11 possible transitive… read more here.
Keywords: galois groups; compositional nonic; mathrm gal; groups power ... See more keywords
WILD RAMIFICATION IN TRINOMIAL EXTENSIONS AND GALOIS GROUPS
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DOI: 10.1017/s001708952000004x
Abstract: Abstract It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the… read more here.
Keywords: wild ramification; trinomial extensions; galois groups; ramification trinomial ... See more keywords
Abelian absolute Galois groups
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DOI: 10.1017/s0017089524000028
Abstract: Abstract Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if $K$ is a finitely generated extension of transcendence degree $r$ of a global field and $A$ is a closed abelian subgroup… read more here.
Keywords: galois groups; absolute galois; abelian absolute;
Infinite families of monogenic trinomials and their Galois groups
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DOI: 10.1142/s0129167x18500398
Abstract: Let n ∈ ℤ with n ≥ 3. Let Sn and An denote, respectively, the symmetric group and alternating group on n letters. Let m be an indeterminate, and define fm(x) := xn + a(m,n)x… read more here.
Keywords: infinite families; families monogenic; galois groups; trinomials galois ... See more keywords
Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
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DOI: 10.1515/forum-2023-0227
Abstract: Abstract Let p be a prime number. One of the most difficult and important questions in Galois theory these days is determining which pro-p groups can occur as maximal pro-p Galois groups. Some restrictions of… read more here.
Keywords: galois groups; pro galois; pro groups; profinite completion ... See more keywords