Let K/F be a separable extension. (i) If K = F(α) with αⁿ ∈ F for some n, K/F is said to be a radical extension. (ii) If there exists a sequence of fields F = F₀ ⊆ F₁ ⊆ ... ⊆ F(s) = K so that Fᵢ₊₁ = Fᵢ(αᵢ) with αᵢⁿ⁽ⁱ⁾ ∈ Fᵢ for some nᵢ ∈ N, charF ∧nᵢ for every i, and [Fᵢ₊₁ : Fᵢ] = nᵢ, K/F is said to be a radical tower. In the first part of this work, we present two theorems which give sufficient conditions for a field extension K/F to be radical. In the second part, we present results which provide conditions under which every subfield of a radical tower is also a radical tower.

Thesis (M.Sc.) -- Dept. of Pure Mathematics, University of Adelaide1983. / Typescript (photocopy).

Thesis (M.S.)--University of North Carolina at Wilmington, 2005. / Includes bibliographical references (Leaf: [104])

Thesis (Doctoral)--Universität Bonn, 1991. / Includes bibliographical references.

Hopf-Galois extensions were introduced by Chase and Sweedler [CS69] in 1969, motivated by the problem of formulating an analogue of Galois theory for inseparable extensions. Their approach shed a new light on separable extensions. Later in 1987, the concept of Hopf-Galois theory was further developed by Greither and Pareigis [GP87]. So, as a problem in the theory of groups, they explained the problem of finding all Hopf-Galois structures on a finite separable extension of fields. After that, many results on Hopf-Galois structures were obtained by N. Byott, T. Crespo, S. Carnahan, L. Childs, and T. Kohl. In this thesis, we consider Hopf-Galois structures on Galois extensions of squarefree degree n. We first determine the number of isomorphism classes of groups G of order n whose centre and commutator subgroup have given orders, and we describe Aut(G) for each such G. By investigating regular cyclic subgroups in Hol(G), we enumerate the Hopf-Galois structures of type G on a cyclic extension of fields L/K of degree n. We then determine the total number of Hopf-Galois structures on L/K. Finally, we examine Hopf-Galois structures on a Galois extension L/K with arbitrary Galois group Gamma of order n, and give a formula for the number of Hopf-Galois structures on L/K of a given type G.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 112-116). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--Oregon State University, 2003. / Typescript (photocopy). Includes bibliographical references (leaves 43-46). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Washington, 1996. / Vita. Includes bibliographical references (leaves [47]-48).

No description available.

Let F/F^+ be a quadratic imaginary field extension of a totally real field F^+, and pi cong \tilde{\pi} otimes xi be a cuspidal automorphic representation of GL_n(AA_F) obtained from tilde{pi} by twisting a Hecke character xi. In the case of F^+ = QQ, Michael Harris defined arithmetic automorphic periods for certain tilde{pi} in his Crelle paper 1997, and showed that critical values of automorphic L-functions for pi can be interpreted in terms of these arithmetic automorphic periods. Lin Jie generalized his construction and results to the general totally real field F^+ in her thesis. On the other hand, for certain cuspidal representation Pi of GL_{2n}(F^+), which admits a Shalika model, Grobner and Raghuram related their critical values of L-functions to a non-zero complex number (called Shalika periods). We noticed that the automorphic induction AI(pi) of pi, considered by Harris and Lin, will automatically have a Shalika model, and by comparing common critical values of their identical L-functions, we relate the Shalika periods of AI(pi) with arithmetic automorphic periods of tilde{pi}. In the case F^+=QQ, this comparison will express each arithmetic automorphic period in terms of the corresponding Shalika periods.