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Showing papers on "Generic polynomial published in 1999"


Journal ArticleDOI
TL;DR: The existence of an element w in E satisfying the following conditions is proved, which establishes a recent conjecture of Morgan and Mullen, who, by means of a computer search, have verified the existence of such elements for the cases in which q≤ 97 and n≤ 6, n being the degree of E over F.
Abstract: Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let a∈F be nonzero. We prove the existence of an element w in E satisfying the following conditions: - w is primitive in E, i.e., w generates the multiplicative group of E (as a module over the ring of integers). - the set {w g ∣g∈G} of conjugates of w under G forms a normal basis of E over F. - the (E, F)-trace of w is equal to a. This result is a strengthening of the primitive normal basis theorem of Lenstra and Schoof [10] and the theorem of Cohen on primitive elements with prescribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of such elements for the cases in which q≤ 97 and n≤ 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically.

41 citations


Book ChapterDOI
01 Jan 1999
TL;DR: In this paper, it was shown that for any prime number l ≥ 19, the l-primary part of a point on the modular curve P lies in the cuspidal subgroup C of J0(N).
Abstract: Suppose that N is a prime number greater than 19 and that P is a point on the modular curve X0(N) whose image in Jo(N) (under the standard embedding ι: X0(N)→J0(N)) has finite order. In [2], Coleman-Kaskel-Ribet conjecture that either P is a hyperelliptic branch point of X0(N) (so that N∈{23,29,31,41,47,59,71}) or else that ι(P) lies in the cuspidal subgroup C of J0(N). That article suggests a strategy for the proof: assuming that P is not a hyperelliptic branch point of X0(N), one should show for each prime number l that the l-primary part of ι(P) lies in C. In [2], the strategy is implemented under a variety of hypotheses but little is proved for the primes l=2 and l=3. Here I prove the desired statement for l=2 whenever N is prime to the discriminant of the ring End J0(N). This supplementary hypothesis, while annoying, seems to be a mild one; according to W.A. Stein of Berkeley, California, in the range N<5021, it is false only in case N=389.

19 citations


Patent
02 Feb 1999
TL;DR: In this article, an implementation of a multi-dimensional galois field multiplier and a method of Galois Field Multi-dimensional multiplication which are able to support many communication standards having various symbol sizes (16, different GFs(14), and different primitive polynomials(12), in a cost efficient manner is disclosed.
Abstract: An implementation of a multi-dimensional galois field multiplier and a method of galois field multi-dimensional multiplication which are able to support many communication standards having various symbol sizes(16), different GFs(14), and different primitive polynomials(12), in a cost-efficient manner is disclosed. The key to allow a single implementation to perform for all different GF sizes is to shift the one of the two operands(16) and primitive polynomial(12) to the left and to shift the intermediate output ZO(28) to the right in dependence upon the relative size of the GF(14) as compared to the size of the operand, primitive polynomial or intermediate output, whichever is being shifted. The shifting of the above-mentioned signals allows the MULT-XOR arrays(26) to operate on all fields with the exact same hardware with a minimum delay of 2 gates per block or with a critical delay of 2 XOR gates.

13 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the torsion in the second Chow group of products of Severi-Brauer varieties and its relationship with the relative Galois cohomology group H3(L/F) for a generic splitting field of a central simple F-algebra.
Abstract: A field extension L / F is called excellent if, for any quadratic form φ over F, the anisotropic part (φL)an of φ over L is defined over F; L / F is called universally excellent if L ⋅ E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras.

11 citations


Journal ArticleDOI
23 Sep 1999
TL;DR: In this article, the Tate-Shafarevich groups of A over K and A over F were derived under certain restrictions on A and K/F. Assuming that these groups are finite, a formula for the order of the subgroup of IU(A/K) of G-invariant elements was derived.
Abstract: Let K/F be a finite Galois extension of number fields with Galois group G, let A be an abelian variety defined over F, and let EU(A/K) and III(A/F) denote, respectively, the Tate-Shafarevich groups of A over K and of A over F. Assuming that these groups are finite, we derive, under certain restrictions on A and K/F, a formula for the order of the subgroup of IU(A/K) of G-invariant elements. As a corollary, we obtain a simple formula relating the orders of IU(A/K), IU(A/F) and 1U(A'F) when K/F is a quadratic extension and AX is the twist of A by the non-trivial character X of G.

11 citations


01 Jan 1999
TL;DR: In this article, explicit formulae for the five roots of DeMoivre's quintic polynomial are given in terms of any two of the roots, and it is known that f(x) is solvable by radicals if and only if all the roots of f (x) can be expressed as rational functions of any 2 of them.
Abstract: Explicit formulae for the five roots of DeMoivre's quintic polynomial are given in terms of any two of the roots. If f(x) is an irreducible polynomial of prime degree over the rational field Q, a classical theorem of Galois asserts that f(x) is solvable by radicals if and only if all the roots of f(x) can be expressed as rational functions of any two of them, see for example [2, p. 2541. It is known that

7 citations


Journal ArticleDOI
TL;DR: In this article, a Galois graph G(Φ(x, y) is defined by taking the algebraic closure as the vertex set and adjacencies corresponding to the zeroes of the symmetric polynomial over a perfect field k of characteristic zero, and some graph properties such as lengths of walks, distances and cycles are described in terms of Φ.
Abstract: Given a symmetric polynomial Φ(x, y) over a perfect field k of characteristic zero, the Galois graph G(Φ) is defined by taking the algebraic closure as the vertex set and adjacencies corresponding to the zeroes of Φ(x, y). Some graph properties of G(Φ), such as lengths of walks, distances and cycles are described in terms of Φ. Symmetry is also considered, relating the Galois group Gal( ) to the automorphism group of certain classes of Galois graphs. Finally, an application concerning modular curves classifying pairs of isogeny elliptic curves is revisited.

6 citations


Patent
12 Nov 1999
TL;DR: In this paper, a Reed-Solomon (RS) decoder for determining roots of error locator polynomials is presented. Butler et al. present a decoder with a first pre-computation operation to obtain a p-bit polynomial solution value in a first clock cycle and second parallel feedback logical operations are performed in each subsequent clock cycle.
Abstract: A system and method used in a Reed-Solomon (RS) decoder for determining roots of error locator polynomials in which a first pre-computation operation is performed to obtain a p-bit polynomial solution value in a first clock cycle and second parallel feedback logical operations are performed to obtain a p-bit polynomial solution value in each subsequent clock cycles. The system excludes constant Galois Field multipliers from the critical timing path of the system so as to facilitate high speed error-locator polynomial root determination. In the case of an unshortened RS(m,d) decoder defined over the Galois Field GF(2p) where GF(2p) is a finite field of 2p elements and m=2p-1, final root location values are obtained in m cycles. In the case of a shortened RS(n,d) decoder defined over the Galois Field GF(2p) where GF(2p) is a finite field of 2p elements and m=2p-1 and n∫m, final root location values are obtained in n cycles.

4 citations


Journal Article
TL;DR: In this article, it was shown how to determine the other three roots of a monic irreducible quintic polynomial in Q[X] with Galois group D5 in accordance with a theorem of Galois.
Abstract: Let r1 and r2 be any two roots of a monic irreducible quintic polynomial in Q[X] with Galois group D5. It is shown how to determine the other three roots as rational functions of r1 and r2 in accordance with a theorem of Galois.

3 citations


Patent
08 Nov 1999
TL;DR: A table matching method for multiplication of elements in Galois field is presented in this paper, where the byte value is the product of the two elements in the Galois Field, and the two exponents are then added up to obtain a sum.
Abstract: A table matching method for multiplication of elements in Galois Field. First, a table of the byte value in Galois Field and the corresponding exponent is formed in the hardware. To perform the multiplication between two elements in the Galois Field, the corresponding exponents of the two elements are found out in advance. The two exponents are then added up to obtain a sum. Then, by using the table, a corresponding byte value of the sum can be obtained. The byte value is the product of the two elements in the Galois Field.

2 citations


Journal ArticleDOI
TL;DR: It is shown that an optimal nested radical with roots of unity for α can be effectively constructed from the derived series of the solvable Galois group of L(ζ n ) over k( ζ n ).
Abstract: Let k be an algebraic number field. Let α be a root of a polynomial f ∈ k[x] which is solvable by radicals. Let L be the splitting field of α over k. Let n be a natural number divisible by the discriminant of the maximal abelian subextension of L, as well as the exponent of G(L/k), the Galois group of L over k. We show that an optimal nested radical with roots of unity for α can be effectively constructed from the derived series of the solvable Galois group of L(ζ n ) over k(ζ n ).

01 Jan 1999
TL;DR: In this paper, conditions for a polynomial which together with its higher derivatives are integer-valued are derived for the elements from the ring of polynomials over a Galois field.
Abstract: Consider the elements from the ring ) (x q F of polynomials over a Galois field q F as integers. A polynomial ) (T f over ) (x q F is said to be integer-valued if ) (T f takes values in ) (x q F for all T in . ) (x q F We derive conditions for a polynomial which together with its higher derivatives are integer-valued.

Posted Content
TL;DR: In this paper, the fundamental group G of the moduli space of u-compatible pairs of complex structures is defined, which is a kernel of several (equivalent) actions of the braid-cyclic group on 2g strands.
Abstract: Let M and N be even-dimensional oriented real manifolds, and $u:M \to N$ be a smooth mapping. A pair of complex structures at M and N is called u-compatible if the mapping u is holomorphic with respect to these structures. The quotient of the space of u-compatible pairs of complex structures by the group of u-equivariant pairs of diffeomorphisms of M and N is called a moduli space of u-equivariant complex structures. The paper contains a description of the fundamental group G of this moduli space in the following case: $N = CP^1, M \subset CP^2$ is a hyperelliptic genus g curve given by the equation $y^2 = Q(x)$ where Q is a generic polynomial of degree 2g+1, and $u(x,y) = y^2$. The group G is a kernel of several (equivalent) actions of the braid-cyclic group $BC_{2g}$ on 2g strands. These are: an action on the set of trees with 2g numbered edges, an action on the set of all splittings of a (4g+2)-gon into numbered nonintersecting quadrangles, and an action on a certain set of subgroups of the free group with 2g generators. $G_{2g} \subset BC_{2g}$ is a subgroup of the index $(2g+1)^{2g-2}$. Key words: Teichm\"uller spaces, Lyashko-Looijenga map, braid group.