Fix a prime number p, a number field K, and a finite set S of primes of K. Let Sp be the set of all primes of K of residue characteristic p. Inside a fixed algebraic closure K of K, let KS be the maximal p-extension (Galois extension with pro-p Galois group) of K unramified outside S, and put GS = Gal(KS/K). The study of these “fundamental groups” is governed by a dichotomy between the tame (S∩...

In this article we prove an analogue of Bauer’s theorem from algebraic number theory in the context of hyperbolic systems. 0. Introduction Let φ : X → X be a hyperbolic dynamical system and, for a finite G, let φ̃ : X̃ → X̃ be a G-extension for which φ̃ is also hyperbolic. Given φ we are interested in describing the possibilities for φ̃ in terms of the closed φ-orbits. This is analogous to a classic...

In this paper a duality principle is formulated for statements about skew field extensions of finite (left or right) degree. A proof for this duality principle is given by constructing for every extension L/K of finite degree a dual extension LJK, . These dual extensions are constructed by embedding a given L/K in an inner Galois extension N/K. The Appendix shows that such an embedding can alwa...

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Zp-extension of the CM field, where p is a prime of good, ordinary reduction for E. When the complex L-function of E vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the p-power Selmer group over the anticyclotomic extension is a torsion Iwasa...

We consider finite subsets Λ ⊂ Zn satisfying the extension property, i.e. the property that every collection {ck}k∈Λ−Λ of complex numbers which is positive-definite on Λ is the restriction to Λ − Λ of the Fourier coefficients of some positive measure on Tn. A simple algebraic condition on the set of trigonometric polynomials with non-zero coefficients restricted to Λ is shown to imply the failu...

Let k be a field and V be an algebraic fc-scheme of dimension w—1. V is called a Severi-Brauer /;-scheme if there exists a separable algebraic extension L/k such that V xkL and Pw_1(L)=Proj L[XX, • • • , Xn] are isomorphic as /.-schemes [7]. V is said to be split by L/k. V is called a trivial Severi-Brauer ^-scheme if V and -Pw_i(fc) are isomorphic as fcschemes. Let K/k be a finite Galois exten...

Let E/Q be an elliptic curve defined over Q of conductor N and let Gal(Q/Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism σ ∈ Gal(Q/Q), we let Q be the fixed subfield of Q under σ. We prove that for every σ ∈ Gal(Q/Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in Q σ has infinite rank, so the rank of E(Q σ ) is infinite. Our ...

An extension (K(X)|K, v) of valued fields is said to be valuation transcendental if we have equality in the Abhyankar inequality. Minimal pairs definition are fundamental objects investigation extensions. In this article, associate a uniquely determined positive integer with extension. This defined via chosen minimal pair definition, but it later shown independent choice. Further, show that enc...

the purpose of this study was to assess the process of extension program development in iranian agricultural extension system (iaes). the research instrument was a structural questionnaire with close-ended questions which its validity and reliability was confirmed by using expert panel and cranach’s alpha test, respectively. the target population of this research included all extension managers...

Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that TrF/E(g(x), h(x)) = δg,h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char(E) 6= 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char(E) = 2, then F ...