in a previous paper, the second author established that, given finite fields $f < e$ and certain subgroups $c leq e^times$, there is a galois connection between the intermediate field lattice ${l mid f leq l leq e}$ and $c$'s subgroup lattice. based on the galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product $c rtimes {rm gal} (e/f)$...

We study the question of surjectivity Galois correspondence from subHopf algebras to subfields given by Fundamental Theorem Theory for abelian Hopf structures on a extension fields with group G', finite p-group. Applying connection between regular subgroups holomorph p-group (G, +) and associative, commutative nilpotent algebra A Caranti, et. al., we show that if gives rise H-Hopf structure L/K...

In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that ...

Example 1.1. Two R-automorphisms of C are the identity z 7→ z and complex conjugation z 7→ z. We will show they are the only ones. If σ : C→ C is an R-automorphism, then for any real a and b we have σ(a+ bi) = σ(a) +σ(b)σ(i) = a+ bσ(i), so σ is determined by σ(i) and i = −1 =⇒ σ(i) = σ(−1) =⇒ σ(i) = −1 =⇒ σ(i) = ±i. If σ(i) = i, then σ(z) = z for all z ∈ C and if σ(i) = −i, then σ(z) = z for al...

We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group Γ, a finite abelian p-group. Applying the connection between regular subgroups of the holomorph of a finite abelian p-group (G,+) and associative, commutati...

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in proof Mordell–Lang conjecture, we describe a correspondence between action group and dynamical rational map. For nonlinear polynomials coefficients, irreducibility associated dynatomic polynomial serves as convenient criterion, although also verify occurs several cases when is reducible. Th...

Any σ ∈ Gal(L1L2/K) restricted to L1 or L2 is an automorphism since L1 and L2 are both Galois over K. So we get a function R : Gal(L1L2/K)→ Gal(L1/K)×Gal(L2/K) by R(σ) = (σ|L1 , σ|L2). We will show R is an injective homomorphism. To show R is a homomorphism, it suffices to check the separate restriction maps σ 7→ σ|L1 and σ 7→ σ|L2 are each homomorphisms from Gal(L1L2/K) to Gal(L1/K) and Gal(L2...