Showing papers on "Generic polynomial published in 1989"

An elementary test for the Galois group of a quartic polynomial

Abstract: (1989). An Elementary Test for the Galois Group of a Quartic Polynomial. The American Mathematical Monthly: Vol. 96, No. 2, pp. 133-137.

Multiplier-adder in the Galois fields, and its use in a digital signal processing processor

Abstract: The multiplier-adder in the Galois fields can have parameters applied to it, i.e. it is possible to choose the Galois field CG(2 m ) in which the polynomial operations are performed, with m at most equal to N, N being predetermined by the designer. The multiplier-adder is made up of a decoder (10) organized as N identical elementary cells receiving the generator polynomial G(m:0) and supplying the generator polynomial without its least significant bit G(m-1:0) and a polynomial marking the degree of the generator polynomial, DG(m-1:0), and a computing matrix (20) organized as N columns of identical elementary cells receiving the polynomials A, B and C of the Galois field CG(2 m ) and supplying a polynomial result P=(A*B) modulo G +C. The multiplier-adder has usage for example as a digital signal processing processors for error detecting and correcting encoding and decoding using BCH or RS codes.

Galois groups and factoring polynomials over finite fields

Abstract: Let p be a prime and F be a polynomial with integer coefficients. Suppose that the discriminant of F is not divisible by p, and denote by m the degree of the splitting field of F over Q and by L the maximal size of the coefficients of F. Then, assuming the generalized Riemann hypothesis (GRH), it is shown that the irreducible factors of F modulo p can be found in deterministic time polynomial in deg F, m, log p, and L. As an application, it is shown that it is possible under GRH to solve certain equations of the form nP=R, where R is a given and P is an unknown point of an elliptic curve defined over GF(p) in polynomial time (n is counted in unary). An elliptic analog of results obtained recently about factoring polynomials with the help of smooth multiplicative subgroups of finite field is proved. >

On the Galois groups of the iterates of x 2 +1

Abstract: §1. Introduction . In [1], Odoni discusses the iterates of the polynomial x 2 +1 and their Galois groups over the rationals (a problem initially proposed by J. McKay). Setting f 1 ,( x ) = x 2 +1 and f n ( x ) = f 1 (f n-1 ( X )) for n ≥ 2, write K n for the splitting field of f n ( x ) over and Ω n = Gal ( K n / ). Then Odoni proves that Ω n is isomorphic to a subgroup of [C 2 ] n , the n th wreath power of the cyclic group C 2 of order 2, and gives a simple rational criterion for Ω n [C 2 ] n to hold. In this note we describe a computer implementation of Odoni's criterion, and state the result that Ω n [C 2 ] n for n ≤ 5 × 10 7 .

AT/sup 2/-optimal Galois field multiplier for VLSI

Abstract: VLSI designs for Galois field multipliers, which are central in many encoding and decoding procedures for error-detecting and error-correcting codes, are presented. An AT/sup 2/-optimal Galois-field multiplier based on AT/sup 2/-optimal integer multipliers for a synchronous VLSI model is exhibited. Galois field multiplication is done in two steps. First two polynomials (of degree n-1) over Z/sub p/ are multiplied, and then the resulting polynomial is reduced modulo a fixed irreducible polynomial (of degree n). Multiplication of polynomials is done by discrete Fourier transform (DFT). For p=2, the procedure is more involved for Z/sub p/(x) than for Z(x). An extension to the case of variable p is included and some open problems are stated. >

Representation of a real polynomial f(X) as a sum of 2m-th powers of rational functions

Abstract: From Becker’s Satz 2.14 in [B1] it follows that a polynomial f ∈ ℝ[X] admits a representation $$ f = \sum\limits_{{i = 1}}^{\sigma } {\frac{{g_i^{{2m}}}}{{{h^{{2m}}}}}} $$ (1) with gi, h∈ ℝ[X] if and only if f satisfies the following three conditions: (i) 2m divides deg f (ii) 2m divides the order of every real zero of f (iii) f is positive semidefinite Once f satisfies these conditions, the problem arises how to obtain a representation (1) for f. This paper is concerned with that problem.

Embedding problems with ramification conditions

Abstract: An interesting point concerning embedding problems over number fields is the knowledge of the ramification set of its solutions. In the present work, we examine the following problem: Let K be a number field, G K its absolute Galois group, ~b an epimorphism from G K onto a finite group G and L[ K the Galois extension associated to qS. We consider the embedding problem: G~ 1 ~A >E ~G ~1 where E is a central extension of G, i.e. A is a trivial G-module, and assume ~b*e = 0 in H2(GK, A), for e the element in H2(G, A) corresponding to E. For the solvable embedding problem (L J K, e), we want to get a solution field M such that the extension M [ K has a reduced ramification set. To study this problem, we take a finite set S of prime ideals of the ring of integers (9 K of the field K containing the prime ideals ramifying in L[ K. We state the embedding problem (L I K, 0 with the additional condition that the solutions are unramified outside S. We denote this new problem by (L I K, e, S). If G s is the Galois group of the maximal extension of K, unramified outside S, we have a commutative diagram:

Resolvents and symmetric functions

Abstract: In the present paper a model of transformations of polynomial equations (the so called “direct image” model) is studied. We express, in this model, some minimal polynomials and some resolvents relative to the Galois group of a polynomial in order to use a general algorithm of resolution. This algorithm can be effectively computed in MACSYMA with the extension SYM that manipulates symmetric polynomials. We give a few examples obtained by specializing the general algorithm for the Galois resolvent.

Adjustable-parameter multiplier/summer in galois bodies, and its use in a digital signal processor

Abstract: The multiplier/summer in the Galois bodies is parametrisable, i.e it is possible to chose the Galois body CG(2m) in which the polynomial operations are performed, with m being at the most equal to N, N being predetermined by the designer. It comprises: a decoder (10) organized into N identical elementary cells which receive the generating polynomial G(m:0), and supplying the generating polynomial without its low-weight bit, G(m-1:0) and a polynomial marking the degree of the generating polynomial DG(m-1:0), and a calculating matrix (20), organized into N columns of identical elementary cells, receiving the polynomials A, B and C of the Galois body CG(2m) and supplying a polynomial result P = (A*B)modulo G+C. The invention applies to digital signal processors, for error detecting and correcting coding and decoding systems using BCH or RS codes.

Rationality Criteria for Galois Extensions

Abstract: Some rationality criteria for finite Galois extensions over C(t) are explained. The first rationality criterion and the second rationality criterion, together with the corresponding examples are contained in the forthcoming lecture notes (27) (see also (23-25)). The rationality criteria in sections 4 and 5, the braid orbit theorem, and the twisted braid orbit theorem, are new. With the last one, the Mathieu group M24 is realized as Galois group over Q. More than 150 years ago Galois attached to every polynomial (over a given field) without double zeros a finite group. In principle he proceeded in the following way: Let K be a field, K an algebraic closure of K and f(X) E K(X) a separable polynomial of degree m with zeros (h, ... , 8m in K. Then the set of polynomials in K(X) := K(X1"" ,Xm), defined by