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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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TL;DR: For a transitive subgroup G ≥ S 6 which contains C 3 × C 3 as subgroup, this article proved that G is rational over K, where K is any field, and G acts naturally on K(x 1,…, x 6) by permutations on the variables.
Abstract: For a transitive subgroup G ≤ S 6 which contain C 3 × C 3 as subgroup, we prove that K(x 1,…, x 6) G is rational over K, where K is any field, and G acts naturally on K(x 1,…, x 6) by permutations on the variables. We also give an application on construction of generic polynomials.
3 citations
TL;DR: For a cyclic Galois extension M = L(α 1/n ) of L of degree n such that M is Galois over F, this article gave an explicit parametrization of those a that lead to each possible group Gal(M/F).
Abstract: Let n be any integer with n > 1, and let F C L be fields such that [L: F] = 2, L is Galois over F, and L contains a primitive n th root of unity ζ. For a cyclic Galois extension M = L(α 1/n ) of L of degree n such that M is Galois over F, we determine, in terms of the action of Gal(L/F) on a and ζ, what group occurs as Gal(M/F). The general case reduces to that where n = p e , with p prime. For n = p e , we give an explicit parametrization of those a that lead to each possible group Gal(M/F).
3 citations
Patent•
18 Nov 2002
TL;DR: One kind of Galois field multiplier system is the product line converter circuit (PLC) as discussed by the authors, which uses polynomials with coefficients multiplied to obtain the product thereof multiplier circuit.
Abstract: One kind of Galois field multiplier system (10) comprising: means for on the Galois field two polynomials with coefficients multiplied to obtain the product thereof multiplier circuit (12); in response to the multiplier circuit, with the prediction polynomial product line converter circuit (18) for polynomial modulo remainder irreducible Galois field; and means for providing a set of coefficients to the Galois field linear transformer circuit in a prediction of a predetermined irreducible polynomial modulus remainder memory circuit (20).
3 citations
TL;DR: In this article, a weak restriction on the ramification filtration associated with each prime of K above 2 was introduced, and it was shown that the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone.
Abstract: Let NUK be a biquadratic extension of algebraic number fields, and G = Gal(NUK). Under a weak restriction on the ramification filtration associated with each prime of K above 2, we explicitly describe the Z[G]-module structure of each ambiguous ideal of N. We find under this restriction that in the representation of each ambiguous ideal as a Z[G]-module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone. For a given group, G, define SG to be the set of indecomposable Z[G]-modules, M , such that there is an extension, NUK, for which G ≤ Gal(NUK), and M is a Z[G]-module summand of an ambiguous ideal of N. Can SG ever be infinite? In this paper we answer this question of Chinburg in the affirmative. Received by the editors April 21, 1997. I would like to express my appreciation to Ted Chinburg, Roger Wiegand and Steve Wilson for helpful suggestions, and especially to Jacques Queyrut and Universite de Bordeaux for support during the summer of 1995 while part of this paper was being completed. AMS subject classification: Primary: 11R33; secondary: 11S15, 20C32.
3 citations
TL;DR: The improved Hodge polygon (IHP) as discussed by the authors is a generic polynomial whose convex hull is an $n$-dimensional paralleltope whose Galois group is isomorphic to the Newton polygon.
Abstract: Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline x)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of varieties whose Galois group is isomorphic to $\mathbb{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a finite character of $\mathbb{Z}_p$ and a generic polynomial whose convex hull is an $n$-dimensional paralleltope $\Delta$. We denote this polygon by $\mathrm{GNP}(\Delta)$. We prove a lower bound of $\mathrm{GNP}(\Delta)$, which is called the improved Hodge polygon $\mathrm{IHP}(\Delta)$. We show that $\mathrm{IHP}(\Delta)$ lies above the usual Hodge polygon $\mathrm{HP}(\Delta)$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $\Delta$, it coincides with $\mathrm{GNP}(\Delta)$ at these points. As a corollary, we roughly determine the distribution of the slopes of $\mathrm{GNP}(\Delta)$.
3 citations