Showing papers on "Generic polynomial published in 1996"

Finite fields and their applications

Abstract: This chapter discusses finite fields and their applications. It also traces the historical development of finite fields, outlining some of their basic properties. In general, F q denotes the finite field of order q . The general theory of finite fields may be said to begin with the work of Carl Friedrich Gauss and Evariste Galois. Galois supposes Fx to be irreducible mod p and of degree v and solves Fx ≡ 0 by introducing new symbols, which might be just as useful as the imaginary unit i in analysis. Galois approach via imaginary roots and Dedekind's approach via residue class rings are shown to be essentially equivalent by Kronecker. A careful choice of the representation of a finite field F q may assist in the algorithms for the implementation of arithmetic operations in F q . Enumeration theorems for ordered bases of various types are known. Irreducible polynomials of degree n over F q are important for the construction of the field F q n .

A Geometric Interpretation of the Solution of the General Quartic Polynomial

Abstract: (1996). A Geometric Interpretation of the Solution of the General Quartic Polynomial. The American Mathematical Monthly: Vol. 103, No. 1, pp. 51-57.

Computation of the splitting fields and the Galois groups of polynomials

Abstract: This study is a continuation of Yokoyama et al. [22], which improved the method by Landau and Miller [11] for the determination of solvability of a polynomial over the integers. In both methods, the solvability of a polynomial is reduced, in polynomial time, to that of polynomials, each of which is constructed so that its Galois group acts primitively on its roots. Then, by virtue of Palfy’s bound [14], solvability of polynomials with primitive Galois groups can be determined in polynomial time. An effective method, thus, exists in theory. For practical computation, however, the most serious problem remains: How to determine solvability of each polynomial with primitive Galois group.

Hopf Galois Kummer theory of formal groups

Abstract: Let /: F ?> G be an isogeny between finite n-dimensional formal groups defined over R, the valuation ring of some field extension K of Qp. Let H be the /?-Hopf algebra which arises from this isogeny. For such H, we classify Gal (H), the group of /7-Galois objects. Let Ai be the maximal ideal of R, and let P(F, K) denote the n-tuples of M under the group operation induced by F. Our main result is the construction of an isomorphism from the cokernel of P(f) to Gal (//), where P(f) is the induced map from P(F,K) to P(G,K). In geometric language Gal(//) describes the group of isomorphism classes of principal homogeneous spaces for Spec (//) over Spec (/?). Geometric methods have been used by Mazur to establish the above isomorphism, but the proof is nonconstructive. A geometric approach has also provided a formula for the cardinality of Gal (//). We give an alternative derivation of this result using formal group techniques. Introduction. Given a field K and a finite abelian group G a question in classical field theory is to determine Gal (K, G), the isomorphism classes of Galois field extensions of K with group G. When K has characteristic zero and contains a primitive nth root of unity, classical Kummer theory asserts that Gal (K, Cn) = U(K)/U(K)n where U(K) denotes the units of K and Cn is the cyclic group of order n. Furthermore, there is an explicit isomorphism from U(K)/U(K)n to Gal(/_\ Cn) given by sending u G U(K) (mod U(K)n) to K(x)/(xn - u). In their development of Hopf Galois theory Chase and Sweedier (1) provide criteria for an action (or co-action) of a cocommutative Hopf algebra on an exten sion of rings to be Galois. An /.-algebra S which admits a Galois co-action by an /.-Hopf algebra H is called an //-Galois object. Given a ring R and a cocommu tative /.-Hopf algebra //, a basic question in this theory is to determine Gal (//), the group of isomorphism classes of //-Galois objects. When K is a field and H is the dual of the group ring KG, G a finite group, Gal (//) becomes Gal (K, G). Determining Gal (//) is also of great interest because in geometric language this describes the group of isomorphism classes of principal homogeneous spaces for Spec (//) over Spec (/.). We make the conventions that all rings are commutative, all formal groups are commutative, and all Hopf algebras are abelian (commutative and cocommu tative). An /.-Hopf algebra is called finite if it is finitely generated and projective as an /.-module.

Composita of sextic fields, theory and examples

Abstract: Fix a ground field F. Let be a finite collection of finite degree separable field extensions of F. To study how the fields in К are related to each other, one should construct a joint splitting field F К and describe the Galois group Gal(F К /F) In this paper we describe the general appearance of Gal(F К /F) for arbi-trary F when the F i all have degree ≤ 6. Then we apply this description to three interesting related examples. First and second, we take ground fields the p-adic fields Q 2 and Q 3, and К the collection of all degree ≤ 6 fields. Third, we take ground field the rational number field Q and К the collection of all degree ≤ 6 fields with absolute discriminant of the form 2a3b This paper complements three of our previous papers. The paper [4] roughly speaking considers the Galois theory associated to a single field K of degree ≤ 6, whereas here we consider a finite collection of such K. The paper [5] lists all degree ≤ 6 fields over Q 2 and Q 3 . These tables form the basis of our local examples....