Showing papers on "Generic polynomial published in 1998"

Galois Theory for Minors of Finite Functions

Abstract: Galois Theory for Minors of Finite Functions Nicholas Pippenger A Boolean function f is a minor of a Boolean function g if f is obtained from g by substituting an argument of f, the complement of an argument of f, or a Boolean constant for each argument of g. The theory of minors has been used to study threshold functions (also known as linearly separable functions) and their generalization to functions of bounded order (where the degree of the separating polynomial is bounded, but may be greater than one). We construct a Galois theory for sets of Boolean functions closed under taking minors, as well as for a number of generalizations of this situation. In this Galois theory we take as the dual objects certain pairs of relations that we call ``constraints'''', and we explicitly determine the closure conditions on sets of constraints.

Fast inversion in composite Galois fields GF((2/sup n/)/sup m/)

Abstract: We describe an improvement of Itoh and Tsujii's algorithm for inversion over Galois fields GF((2/sup n/)/sup m/). In particular, raising an element to the 2/sup ln/ power, l an integer, in polynomial basis representation can be done with a binary, fixed matrix. Finally, we show that the inversion complexity is essentially given by the number of multiplications.

Galois module structure of ambiguous ideals in biquadratic extensions

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.

Topological models and cohomology of Galois groups

Abstract: Let F denote a field of characteristic different from 2. In this Note we describe the mod 2 cohomology of a Galois group ς F which is determined by the Witt ring W F .

On a conjecture of Le Bruyn

Abstract: Given a generic field extension F/k of degree n>3 (i.e. the Galois group of the normal closure of F is isomorphic to the symmetric group $S_n$), we prove that the norm torus, defined as the kernel of the norm map $N:R_{F/k}(G_m)\to\G_m$, is not rational over k.

On the galois group of f(xd )

Abstract: Let K be a field which contains the d-th roots of unity. Suppose K has characteristic 0 or char(K) ∤ d. Let f(X) ∈ K[X] be an irreducible and separable polynomial of degree m with Galois group 𝔤. If 𝔤 acts doubly transitively on the set of roots of f(X d ) and if the order of 𝔤 and d are relatively prime, then the Galois group of f(X d ) over a splitting field of f(X) is isomorphic to where d 3 | d 2 | d 1 | d. We prove that for the Mathieu groups and some other groups the same statement holds true for any . We further give equivalent interpretations for d 1 and d 2.