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Abstract: This paper provides a method for identifying “sufficiently rich” simple groups of finite Morley rank with simple algebraic groups over algebraically closed fields. Special attention is given to the even type case, and the paper contains a number of structural results about simple groups of finite Morley rank and even type. [Copyright &y& Elsevier]

The notion of a ∗-ordering on a skew-field with involution was introduced by S. Holland in [J. Algebra 101 (1986) 16–46] as an analogue to the notion of a total ordering on a skew-field and developed further in a series of papers of T. Craven, I. Idris, M. Marshall and T. Smith. While it is well known that every free skew-field has a total ordering (see [J. Lewin, Trans. Amer. Math. Soc. 192 (1974) 339–346]), it has not been known so far whether every free skew-field with some natural involution has a ∗-ordering. The aim of this paper is to give an explicit construction of a class of ∗-orderings on free associative algebras and to prove that ∗-orderings from this class extend uniquely to the corresponding free skew fields. The problem was posed by T. Craven and T. Smith in [J. Algebra 238 (2001) 314–327]. [Copyright &y& Elsevier]

The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin–Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate rotated versions of the densest known lattices in dimensions 2, 3, 4, 5, 6, 7, 8, 12 and 24 constructed via ideals and free Z -modules that are not ideals in subfields of cyclotomic fields. The focus is on totally real number fields and the associated full diversity lattices which may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. We also discuss on the existence of a number field K such that it is possible to obtain the lattices A 2 , E 6 and E 7 via a twisted embedding applied to a fractional ideal of O K . [ABSTRACT FROM AUTHOR]

Abstract: Let K be a complete discrete valuation field of mixed characteristic with residue field such that . Let be the absolute Galois group of K and a p-adic representation. When is perfect, Shankar Sen described the Lie algebra of in terms of so-called Sen's operator Θ for ρ. When may not be perfect, Olivier Brinon defined operators for ρ, which reduce to Sen's operator Θ in the case of . In this paper, we describe the Lie algebra of in terms of Brinon's operators , which is a generalization of Sen's result. [Copyright &y& Elsevier]

Abstract: Let K be a perfect field of characteristic not equal to two, an algebraic closure of K and let be the Galois group of the extension . Let T be an n-point set in . The field of moduli of T is contained in each field of definition but it is not necessarily a field of definition. In this paper we show that point sets of odd cardinality in with field of moduli K are defined over their field of moduli. We, also, show that, except for the special case of the 4-point sets, this does not hold in general for point sets of even cardinality . Finally we prove that the following local-to-global principle holds for point sets of cardinality : if an n-point set T in is defined over for each prime p (including the prime at ∞), then it is defined over . From this result we derive an analogue local-to-global principle for hyperelliptic curves. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to introduce a new class of Noetherian rings of prime characteristic via perfect closure and study their basic properties. If the perfect closure of a Noetherian ring is coherent, we call it an F-coherent ring. Some applications are given to tight closure theory. In particular, we discuss some relationship between F-coherent rings and F-pure, F-regular, and F-injective rings. As a main tool, we use techniques from valuation theory. The final section discusses how the coherent property effects the behavior of tight closure of finitely generated ideals on general perfect rings. [Copyright &y& Elsevier]

Abstract: For any number field, J.-F. Jaulent introduced a new invariant called the group of logarithmic classes in 1994. This invariant is proved to be closely related to the wild kernels of number fields. In this paper, we show how to compute the kernel of the natural homomorphism from the group of logarithmic classes to the group of p-ideal classes by computing the p-adic regulator which is a classical invariant in number theory. As an application, we prove Gangl''s conjecture on 9-rank of the tame kernel of imaginary quadratic field . [Copyright &y& Elsevier]

Abstract: Let p be an odd prime and let F be arbitrary field of characteristic not p, containing a primitive pth root of unity ζ. In this paper, we prove a criterion, giving the obstructions to realizability of p-groups as Galois groups over F, having a factor-group of the kind . We apply this to the non-abelian groups of orders and . Where it is possible, we give a description of all Galois extensions realizing these groups. We discuss also automatic realizations and local fields. [Copyright &y& Elsevier]

Abstract: A Riemann surface X is said to be of type if its full automorphism group AutX is cyclic of order n and the quotient surface has genus m. In this paper we determine necessary and sufficient conditions on the integers and γ, where n is odd, for the existence of a Riemann surface of genus g and type admitting a symmetry with γ ovals. [Copyright &y& Elsevier]

Abstract: Let R be a regular noetherian local ring and let V be the ring of a zero-dimensional valuation v of the quotient field of R dominating R. The aim of this paper is to characterize when V can be obtained from R by the sequence of successive quadratic transforms of R along V. We also show some properties of such valuations in terms of the sequence. [Copyright &y& Elsevier]

Let λ be a 2-regular partition of n into two parts and let Dλ denote the corresponding irreducible F2Σn-module. We say that Dλ is of quadratic type if there is a non-degenerate Σn-invariant quadratic form defined on this module. In this paper, we show that Dλ is not of quadratic type precisely when the smaller part of λ is a power of 2, say 2r, where r⩾0, and n≡kmod2r+2, where k is one of the 2r consecutive integers 2r+1+2r-1, …, 2r+2-2. [Copyright &y& Elsevier]

Let K/k be a cyclic totally ramified Kummer extension of degree pn with Galois group G. Let O and o be the rings of integers in K and k, respectively. For n=1, F. Bertrandias and M.-J. Ferton determined the ring EndoG(O) of oG-endomorphisms of O and L.N. Childs constructed the maximal order S in O whose ring of oG-endomorphisms is a Hopf order. A Hopf order whose linear dual is a Larson order in kG is called a dual Larson order. In this paper, the generators of a dual Larson order are given. We determine EndoG(O) and obtain a maximal dual Larson order contained in EndoG(O). We construct a Hopf Galois extension S in O, whose ring H(S) of oG-endomorphisms is a dual Larson order. We affirm an order S′ in O with a dual Larson order H(S′) is a tame H(S′)-extension and show that S is maximal in such orders S′. [Copyright &y& Elsevier]

The aim of this paper is to give a characterisation of the determinants and signatures of integral ideal lattices over a given algebraic number field. This is then used to obtain an existence criterion for automorphisms of given characteristic polynomial. In particular, we give a different proof of a result of B. Gross and C. McMullen [in preparation]. [Copyright &y& Elsevier]

The first major study of p-groups of maximal class was made by Blackburn in 1958. He showed that an important invariant of these groups is the ‘degree of commutativity.’ Recently (1995) Ferna´ndez-Alcober proved a best possible inequality for the degree of commutativity in terms of the order of the group. Recent computations for primes up to 43 show that sharper results can be obtained when an additional invariant is considered. A series of conjectures about this for all primes have been recorded in [A. Vera-Lo´pez et al., preprint]. In this paper, we prove two of these conjectures. [Copyright &y& Elsevier]