Showing papers on "Generic polynomial published in 1981"
TL;DR: In this article, the authors define the 4-powers4 (k) of a field k to be the least positive integers for which the equation −1=a14+n+n +n+1+n is solvable ink (if at all).
Abstract: Define the 4-powers4 (k) of a fieldk to be the least positive integers for which the equation −1=a14+…+as4 is solvable ink (if at all). We determines4 (k) whenk is a Galois field and prove some results abouts4 (k) whenk is an imaginary quadratic field.
6 citations
4 citations
TL;DR: This work proposes a method to reduce a set of functions to one polynomial of one variable on GF (2 N ) (extension field overGF (2)), which has remarkable properties based on Frobenius transforms.
Abstract: Every digital information processing is essentially represented by a set n functions of m variables on {0, 1}. We propose a method to reduce such a set of functions to one polynomial of one variable on GF (2 N ) (extension field over GF (2)). Such polynomials have remarkable properties based on Frobenius transforms, which are to serve for effective designs and productions of switching circuits.
3 citations
TL;DR: This correspondence describes a method for achieving synthesis of finite state algorithms by the use of a set of logic elements that execute field operations from the Galois field GF[pn].
Abstract: This correspondence describes a method for achieving synthesis of finite state algorithms by the use of a set of logic elements that execute field operations from the Galois field GF[pn]. The method begins with a definition of the algorithm to be synthesized in a completely specified finite state flow table form. A polynomial expansion of this flow table function is derived. A canonical sequential circuit corresponding to this polynomial expansion is defined. Subsequently, the given algorithm is synthesized using the canonical circuit by specification of a number of arbitrary constants in the canonical circuit. A mechanical method for deriving constants used in the canonical circuits is given. Finally, some estimates on complexity of the given circuit structure are stated assuming the most fundamental logic element structures.
3 citations
01 Feb 1981
TL;DR: In this article, it was shown that the exceptional case of the previous theorem cannot hold if -1 is a sum of two squares in F, except possibly when p = 2 and VIT a F.
Abstract: For F a field, G a finite group of exponent n and X an irreducible character of G, we let mF(X) denote the Schur index of X over F. A famous theorem of Brauer states that mF(X) = 1 if ,n Ee F where i'n denotes a primitive nth root of unity. In 1975, Goldschmidt and Isaacs [3] showed that p I mF(X) if p is a prime with the property that thep-Sylow subgroup of Gal(F(;)jF) is cyclic, except possibly when p = 2 and VIT a F. Fein [2] showed that the exceptional case of the previous theorem cannot hold if -1 is a sum of two squares in F. In this note we strengthen the previous results by showing
1 citations