Generic polynomial

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

Some polynomials over Q ( t ) and their Galois groups

Abstract: Examples of polynomials with Galois group over Q(t) corresponding to every transitive group through degree eight are calculated, constructively demonstrating the existence of an infinity of extensions with each Galois group over Q through degree eight. The methods used, which for the most part have not appeared in print, are briefly discussed.

Extractors for Polynomial Sources over Fields of Constant Order and Small Characteristic

Abstract: A polynomial source of randomness over F n is a random variable X = f(Z) where f is a polynomial map and Z is a random variable distributed uniformly over F r for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the (total) degree D of the map f , and the base-q logarithm of the size of the range of f over inputs in F r , denoted by k. For simplicity we call X a (q; D; k)-source.

Generic Polynomials Constructive Aspects Of The Inverse Galois Problem

Abstract: 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 " poly-nomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polyno-mials 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. published by the press syndicate of the university of cambridge A catalogue record for this book is available from the British Library.

On Hopf Galois structures and complete groups

Abstract: Let L be a Galois extension of K, fields, with Galois group Γ. We obtain two results. First, if Γ = Hol(Zpe ), we determine the number of Hopf Galois structures on L/K where the associated group of the Hopf algebra H is Γ (i.e. L ⊗K H ∼ L(Γ)). Now let p be a safeprime, that is, p is a prime such that q =( p − 1)/2 > 2 is also prime. If L/K is Galois with group Γ = Hol(Zp), p a safeprime, then for every group G of cardinality p(p−1) there is an H-Hopf Galois structure on L/K where the associated group of H is G, and we count the structures.