Showing papers on "Generic polynomial published in 2002"
Book•
09 Dec 2002TL;DR: In this article, a constructive approach to the inverse Galois problem is described, where 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.
Abstract: 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.
178 citations
Patent•
30 Jan 2002TL;DR: A Galois field multiplier system as discussed by the authors includes a multiplier circuit for multiplying two polynomials with coefficients over a Galois Field to obtain their product, a GF linear transformer circuit responsive to the multiplier circuit, and a storage circuit for supplying to the GF a set of coefficients for predicting the modulo remainder for a predetermined irreducible polynomial.
Abstract: A Galois field multiplier system includes a multiplier circuit for multiplying two polynomials with coefficients over a Galois field to obtain their product; a Galois field linear transformer circuit responsive to the multiplier circuit for predicting the modulo remainder of the polynomial product for an irreducible polynomial; and a storage circuit for supplying to the Galois field linear transformer circuit a set of coefficients for predicting the modulo remainder for predetermined irreducible polynomial.
77 citations
TL;DR: The integrability of Richardson's exact solution of a discrete-state BCS model can be recovered as a special case of an algebraic Bethe-ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions as mentioned in this paper.
Abstract: We show in detail how Richardson’s exact solution of a discrete-state BCS ~DBCS! model can be recovered as a special case of an algebraic Bethe-ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin’s K-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities Hi from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the Hi ~which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling! are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson’s equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the Hi’s as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.
58 citations
TL;DR: A linear differential operator is constructed that allows one to calculate the genus of the complex curve defined by P= 0, the absolute factorization of P over the algebraic closure of k0, and information concerning the Galois group ofP over ___ k0(x) as well as overk0 (x).
39 citations
TL;DR: A fast algorithm is presented for determining the linear complexity and the minimal polynomial of a sequence with period 2p/sup n/ over GF (q), where p and q are odd prime, and q is a primitive root (mod p/sup 2/).
Abstract: A fast algorithm is presented for determining the linear complexity and the minimal polynomial of a sequence with period 2p/sup n/ over GF (q), where p and q are odd prime, and q is a primitive root (mod p/sup 2/). The algorithm uses the fact that in this case the factorization of x/sup 2p(n)/-1 is especially simple.
35 citations
Patent•
02 Dec 2002TL;DR: A Galois field multiply/multiply-add/multipliply accumulate system (10) as discussed by the authors includes a multiplier circuit for multiplying two polynomials with coefficients over Galois fields to obtain their product, a Galois-field linear transformer circuit responsive to the multiplier circuit, and a storage circuit for supplying to the GFLT a set of coefficients for predicting the modulo remainder for a predetermined irreducible polynomial.
Abstract: A Galois field multiply/multiply-add/multiply-accumulate system (10) includes a multiplier circuit for multiplying two polynomials with coefficients over a Galois field to obtain their product; a Galois field linear transformer circuit responsive to the multiplier circuit for predicting the modulo remainder of the polynomial product for an irreducible polynomial; storage circuit for supplying to the Galois field linear transformer circuit a set of coefficient for predicting the modulo remainder for a predetermined irreducible polynomial; and a Galois field adder circuit for adding the product of the multiplier circuit with a third polynomial with coefficients over a Galois field for performing the multiplication and add operations in a single cycle.
29 citations
22 May 2002
TL;DR: In this article, it was shown that the jacobian J(C) of a hyperelliptic curve C: y 2 = f(x) has only trivial endomorphisms over an algebraic closure K a of the ground field K if the Galois group Gal(f) of the irreducible polynomial f (x) ∈ K[x] is either the symmetric group S n or the alternating group An.
Abstract: In a previous paper, the author proved that in characteristic zero the jacobian J(C) of a hyperelliptic curve C: y 2 = f(x) has only trivial endomorphisms over an algebraic closure K a of the ground field K if the Galois group Gal(f) of the irreducible polynomial f(x) ∈ K[x] is either the symmetric group S n or the alternating group An. Here n > 4 is the degree of f. In another paper by the author this result was extended to the case of certain smaller Galois groups. In particular, the infinite series n = 2 r + 1, Gal(f) = L 2 (2 r ):= PSL 2 (F 2 r) and n = 2 4r+2 +1, Gal(f) = Sz(2 2r+1 ) were treated. In this paper the case of Gal(f) = U 3 (2 m ):= PSU 3 (F 2 m) and n = 2 3m + I is treated.
15 citations
TL;DR: In this article, it was shown that the endomorphism ring of the Jacobian of a curve y = f(x) contains a maximal commutative subring isomorphic to the ring of algebraic integers of the lth cyclotomic field, where l is an odd prime dividing the degree n of the polynomial f and different from the characteristic of the algebraically closed ground field.
Abstract: We prove that, under certain additional assumptions, the endomorphism ring of the Jacobian of a curve y{sup l}=f(x) contains a maximal commutative subring isomorphic to the ring of algebraic integers of the lth cyclotomic field. Here l is an odd prime dividing the degree n of the polynomial f and different from the characteristic of the algebraically closed ground field; moreover, n{>=}9. The additional assumptions stipulate that all coefficients of f lie in some subfield K over which its (the polynomial's) Galois group coincides with either the full symmetric group S{sub n} or with the alternating group A{sub n}.
15 citations
01 Jan 2002
7 citations
TL;DR: In this article, the principal differential ideals of a polynomial ring in indeterminates with coefficients in the ring of differential polynomials and derivation given by a general element of the Picard-Vessiot extension were characterized.
Abstract: We characterize the principal differential ideals of a polynomial ring in indeterminates with coefficients in the ring of differential polynomials in indeterminates and derivation given by a “general” element of and use this characterization to construct a generic Picard-Vessiot extension for . In the case when the differential base field has finite transcendence degree over its field of constants we provide necessary and sufficient conditions for solving the inverse differential Galois problem for these groups via specialization from our generic extension.
4 citations
Patent•
08 Feb 2002
TL;DR: In this paper, a finite field multiplication of first and second Galois elements having n bit places and belonging to a Galois field GF 2 n described by an irreducible polynomial is performed by forming an intermediate result Z of intermediate sums of partial products of bit width 2n−2 in an addition part of the Galois multiplier, whereby after all XOR's are traversed a result E with n bits is computed.
Abstract: Finite field multiplication of first and second Galois elements having n bit places and belonging to a Galois field GF 2 n described by an irreducible polynomial is performed by forming an intermediate result Z of intermediate sums of partial products of bit width 2n−2 in an addition part of a Galois multiplier. The intermediate result Z is processed in a reduction part of a Galois multiplier by modulo dividing by the irreducible polynomial, whereby after all XOR's are traversed a result E with n bits is computed.
TL;DR: In this article, a 6-term exact sequence of Galois cohomology with cyclotomic coefficients for any finite extension of fields whose Galois group has an exact quadruple of permutational representations over it was constructed.
Abstract: We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term exact sequence of Galois cohomology with cyclotomic coefficients for any finite extension of fields whose Galois group has an exact quadruple of permutational representations over it. Examples include cyclic groups, dihedral groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4. Several exact sequences conjectured by Bloch-Kato, Merkurjev-Tignol, and Kahn are proven in this way. In addition, we introduce a more sophisticated version of the classical argument known as "Bass-Tate lemma". Some results about annihilator ideals in Milnor rings are deduced as corollaries.
01 Jan 2002
TL;DR: In this article, it was shown that there is a relation between planar functions of elementary abelian groups and bent polynomials, and several results concerning them were proved.
Abstract: It is shown that there is a relation between planar functions of elementary abelian groups and bent polynomials. Moreover we prove several results concerning them.
01 Mar 2002
TL;DR: In this paper, a generic polynomial for Mod 2 n+2 with two parameters over the 2 n -th cyclotomic field k has been constructed based on an explicit answer for linear Noether's problem.
Abstract: We construct a generic polynomial for Mod 2 n+2, the modular 2-group of order 2 n+2 , with two parameters over the 2 n -th cyclotomic field k. Our construction is based on an explicit answer for linear Noether's problem. This polynomial, which has a remarkably simple expression, gives every Mod 2 n+2-extension L/K with K ⊃ k, #K = ∞ by specialization of the parameters. Moreover, we derive a new generic polynomial for the cyclic group of order 2 n+1 from our construction.
01 Sep 2002
TL;DR: In this paper, the authors consider the problem of expressing the roots of an irreducible polynomial f in terms of elements of the ground field by rational operations and radicals, and give a practical method for constructing a radical expression of f when f is solvable.
Abstract: We consider a fundamental question of Galois theory: how to express the roots of an irreducible polynomial f , deg(f ) > 4 in terms of elements of the ground field by rational operations and radicals. In general, expressing the roots of f in terms of radicals is impossible when deg(f ) > 4. By Galois theory, however, we can test whether f is solvable by checking solvability of its Galois group. We will give a practical method for constructing a radical expression of the roots of f , when f is solvable, and report its experiment on a real computer.
Patent•
18 Nov 2002
TL;DR: One kind of Galois field multiplier system is the product line converter circuit (PLC) as discussed by the authors, which uses polynomials with coefficients multiplied to obtain the product thereof multiplier circuit.
Abstract: One kind of Galois field multiplier system (10) comprising: means for on the Galois field two polynomials with coefficients multiplied to obtain the product thereof multiplier circuit (12); in response to the multiplier circuit, with the prediction polynomial product line converter circuit (18) for polynomial modulo remainder irreducible Galois field; and means for providing a set of coefficients to the Galois field linear transformer circuit in a prediction of a predetermined irreducible polynomial modulus remainder memory circuit (20).
TL;DR: In this paper, an Azumaya Galois extension of B with Galois group K is characterized by using BK, where BK is a central weakly Galois algebra with G induced by K.
Abstract: Let B be a Galois algebra over a commutative ring R with Galois group G, C the center of B, K={g∈G|g(c)=c for all c∈C}, Jg{b∈B|bx=g(x)b for all x∈B} for each g∈K, and BK=(⊕∑g∈K Jg). Then BK is a central weakly Galois algebra with Galois group induced by K. Moreover, an Azumaya Galois extension B with Galois group K is characterized by using BK.
TL;DR: In this article, a Galois algebra with Galois group G is defined, and a characterization of the Galois extension BeG and B(1 − eG) is given in terms of a minimal idempotent ei.
Abstract: Let B be a Galois algebra with Galois group G, Jg ={ b ∈ B | bx = g(x)b for all x ∈ B} for each g ∈ G, eg the central idempotent such that BJg = Beg ,a ndeK = � g∈K, eg ≠1 eg for a subgroup K of G .T henBeK is a Galois extension with the Galois group G(eK )( ={ g ∈ G | g(eK ) = eK }) containing K and the normalizer N(K)of K in G. An equivalence condition is also given for G(eK ) = N(K) ,a ndBeG is shown to be a direct sum of all Bei generated by a minimal idempotent ei. Moreover, a characterization for a Galois extension B is shown in terms of the Galois extension BeG and B(1 − eG).
10 Jul 2002
TL;DR: This paper is devoted to show how to easily determine a non-trivial element in the centre of the Galois group of an irreducible polynomial in ℤ[x] when it exists and how to deal in an efficient way with solvability by radicals when this element is available.
Abstract: This paper is devoted to show, first, how to easily determine, when it exists, a non-trivial element in the centre of the Galois group of an irreducible polynomial in ℤ[x] and, second, how to deal in an efficient way with solvability by radicals when a non-trivial element in the centre of the Galois group of the considered polynomial is available.
Patent•
19 Dec 2002TL;DR: In this paper, a method for multiplication of two factors from the Galois field GF (2m asterisk p) is presented, whereby each factor can be presented as a vector of p partial blocks with a width of m bits.
Abstract: Method for multiplication of two factors from the Galois field GF (2m asterisk p), whereby each factor can be presented as a vector of p partial blocks with a width of m bits. The method involves selection of a reduction polynomial, multiplicative linking of the partial blocks, accumulation of an intermediate result with a reduction of the accumulated intermediate result after each multiplicative linking. An Independent claim is made for a second method for multiplication of two factors from the Galois field.