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Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.


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01 Jan 2011
TL;DR: In this article, a Galois extension of B G with an inner Galois group G is presented, where G = {gi |gi(x )= UixU −1 i for some Ui ∈ B and for all x ∈ G, i = 1,2, ···,n for some integer n invertible in B}.
Abstract: Let B be a Galois extension of B G with an inner Galois group G where G = {gi |gi(x )= UixU −1 i for some Ui ∈ B and for all x ∈ B, i =1 ,2, ··· ,nfor some integer n invertible in B}. Then B is a composition of two Galois extensions B ⊃ B K with an inner Abelian Galois group K and B K ⊃ B G with an inner Galois group G/K where K = {g ∈ G |g(Ui )= Ui for each i}. Descriptions of B ⊃ B K and B K ⊃ B G are given.

1 citations

Journal ArticleDOI
TL;DR: In this article, a new proof of self-duality for Selmer groups is given, showing that if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is selfdual.
Abstract: The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

1 citations

Journal ArticleDOI
TL;DR: For a polynomial f(X) we let r(f) denote the number of real roots of f (X) as discussed by the authors, where f is a real number eld and f is an irreducible polynomial in K[X] of odd degree n. This result was generalized by A. Loewy in the following way:
Abstract: 1. Introduction and Loewy’s theorem. By a classical theorem the number of real roots of an irreducible polynomial f(X) of odd prime degree p over a real number eld K is either 1 or p if the Galois group of f(X) over K is solvable. This result was generalized by A. Loewy in the following way: For a polynomial f(X) we let r(f) denote the number of real roots of f(X). Loewy’s theorem. Let K be a real number eld and f(X) an irreducible polynomial in K[X] of odd degree n. If p is the smallest prime divisor of n and the Galois group of f(X) over K is solvable, then r(f) = 1 or n or satises the inequalities p r(f) n p + 1. When the degree of f(X) is a prime number the above theorem is an immediate corollary to the following Galois’ theorem. Let f(X) be an irreducible separable polynomial over a eld K having a solvable Galois group over K. If the degree of f(X) is a prime number, then any two roots of f(X) generate the splitting eld of f(X) over K. Galois’ theorem, which is basically a group-theoretic result, cannot be generalized to yield a proof of Loewy’s theorem. Indeed, for any odd prime number p and any t, 1 t p, there exists an irreducible polynomial f(X) in Q[X] of degree p 2 with solvable Galois group having t roots 1;:::; t

1 citations

Posted Content
TL;DR: In this paper, the authors presented several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors.
Abstract: We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first classify all maximal p-elementary abelian-by-order p quotients of such G_F. In the case p>2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. We then derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of G_F. Finally, we construct a new family of pro-p-groups which are not absolute Galois groups over any field F.

1 citations

Journal ArticleDOI
TL;DR: In this paper, the authors proved the equivalence of the doubling condition for J(x,f) over big balls centered at the origin, boundedness of the multiplicity function N(f,ℝn), the polynomial type of f, and the growth condition for f.
Abstract: Suppose f:ℝn→ℝn is a mapping of K-bounded p-mean distortion for some p>n−1. We prove the equivalence of the following properties of f: the doubling condition for J(x,f) over big balls centered at the origin, the boundedness of the multiplicity function N(f,ℝn), the polynomial type of f, and the polynomial growth condition for f.

1 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20234
20227
20216
202010
20196
20186