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# Generic polynomial

About: Generic polynomial is a(n) research topic. Over the lifetime, 608 publication(s) have been published within this topic receiving 6784 citation(s).

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09 Dec 2002Abstract: This book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is isomorphic to G. Further, if there is such a Galois extension, find an explicit polynomial over K whose Galois group is the prescribed group G. The main theme of the book is an exposition of a family of �generic� polynomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polynomials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of �generic dimension� to address the problem of the smallest number of parameters required by a generic polynomial.

177 citations

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Abstract: We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P (x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over Fp(x) for almost all primes p.

174 citations

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Abstract: Let f(x) = Σaixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for the symmetric group) if it can be obtained from f(x) by a nondegenerate Tschirnhaus transformation. We show that the minimal number dk(n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert‘s 13th problem.

141 citations

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Abstract: A technique is described for the nontentative computer determination of the Galois groups of irreducible polynomials with integer coefficients. The technique for a given polynomial involves finding high-precision approximations to the roots of the poly- nomial, and fixing an ordering for these roots. The roots are then used to create resolvent polynomials of relatively small degree, the linear factors of which determine new orderings for the roots. Sequences of these resolvents isolate the Galois group of the polynomial. Machine implementation of the technique requires the use of multiple-precision integer and multiple-precision real and complex floating-point arithmetic. Using this technique, the writer has developed programs for the determination of the Galois groups of polynomials of degree N _ 7. Two exemplary calculations are given. Introduction. The existence of an algorithm for the determination of Galois groups is nothing new; indeed, the original definition of the Galois group contained, at least implicitly, a technique for its determination, and this technique has been described explicitly by many authors (cf. van der Waerden (8, p. 189)). These sources show that the problem of finding the Galois group of a polynomial p(x) of degree n over a given field K can be reduced to the problem of factoring over K a polynomial of degree n! whose coefficients are symmetric functions of the roots of p(x). In principle, therefore, whenever we have a factoring algorithm over K, we also have a Galois group algorithm. In particular, since Kronecker has described a factoring algorithm for polynomials with rational coefficients, the problem of determining the Galois groups of such polynomials is solved in principle. It is obvious, however, that a procedure which requires the factorization of a polynomial of degree n! is not suited to the uses of mortal men. In the next sections we describe a practical and relatively simple procedure which has been used to develop programs for polynomials of degrees 3 through 7. Restrictions. The algorithm to be described will apply only to irreducible monic polynomials with integer coefficients. Since any polynomial with rational coefficients can easily be transformed into a monic polynomial with integer coefficients equivalent with respect to its Galois group, these latter two adjectives create no genuine restric- tion. The irreducibility restriction is genuine, however. For suppose p(x) = p,(x)p2(x), and suppose K1 and K2 are the splitting fields of P, and p2, respectively. If K1 n K2 = the rationals, then the Galois group of p(x) is the direct sum of the Galois groups of p,(x) and p2(x), and there is no difficulty. If, on the other hand, K1 n K2 is larger than the rationals, then the group of p(x) is not easily determined from those of p,(x) and p2(x) without explicit knowledge of the relations which exist between the

118 citations

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TL;DR: If F is finite then the polynomial f has exactly n rational roots, and it is found the exact number of a[email protected]?F^x such that f has n rational Roots, for each n.

Abstract: We study the polynomial f(x)=x^q^+^1+ax+b over an arbitrary field F of characteristic p, where q is a power of p and ab 0. The polynomial has arisen recently in several different contexts, including the inverse Galois problem, difference sets, and Muller-Cohen-Matthews polynomials in characteristic 2. We prove f has exactly n rational roots, where [email protected]?{0,1,2,Q+1} and [email protected]?GF(q)=GF(Q). If F is finite then we find the exact number of a,[email protected]?F^x such that f has n rational roots, for each n. We also prove many arithmetic properties of f. For example, if F is finite and f has a rational root r, then f has exactly two rational roots if and only if N"F"/"G"F"("Q")(r-1) 1. The techniques rely on a detailed analysis of the splitting field and Galois group, together with frequent use of Hilbert's Theorem 90.

117 citations