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Showing papers on "Generic polynomial published in 2003"


Journal ArticleDOI
TL;DR: In this paper, the Galois theoretic behavior of the p-primary subgroup SelA(F)p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the group G(K/F) is a p-adic Lie group was studied.
Abstract: This paper concerns the Galois theoretic behavior of the p-primary subgroup SelA(F)p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map sK/F′ SelA(F′)p → SelA(K)pG(K/F′) where F′ is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B.Mazurand M. Harris.

63 citations


Journal ArticleDOI
TL;DR: In this paper, the authors examine over arbitrary fields the possible implications among the concepts due to D. Saltman of generic Galois extension, retract rational extension, the lifting property for Galois extensions, the notions due to G. Smith of generic polynomial, and of descent generic Polynomial due to F. DeMeyer.

60 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied Galois covers of the projective line branched at three points with bad reduction to characteristic p, under the condition that p exactly divides the order of the Galois group, and proved that the field of moduli of such a cover is at most tamely ramified at p.
Abstract: We study Galois covers of the projective line branched at three points with bad reduction to characteristic p, under the condition that p exactly divides the order of the Galois group. As an application of our results, we prove that the field of moduli of such a cover is at most tamely ramified at p.

39 citations


Book ChapterDOI
TL;DR: This work explains how fast polynomial arithmetic can be used to speed up the process of solving the equation f(X) = 0, and extends the algorithms to a more general case of extensions that are no longer Galois.
Abstract: Let f(X) be a separable polynomial with coefficients in a field K, generating a field extension M/K. If this extension is Galois with a solvable automorphism group, then the equation f(X) = 0 can be solved by radicals. The first step of the solution consists of splitting the extension M/K into intermediate fields. Such computations are classical, and we explain how fast polynomial arithmetic can be used to speed up the process. Moreover, we extend the algorithms to a more general case of extensions that are no longer Galois. Numerical examples are provided, including results obtained with our implementation for Hilbert class fields of imaginary quadratic fields.

27 citations


Journal ArticleDOI
TL;DR: In this article, the authors define a new kind of digit system in polynomial rings over finite fields, where the ring is a finite field and the digit system is a polynomial in two indeterminates.

20 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the generalized Laguerre polynomial Lm(m)(x) is irreducible for almost all m and has Galois group Am for even m.

18 citations


01 Jan 2003
TL;DR: In this paper, the number of Hopf Galois structures on a Galois extension of K, fields, with Galois group Γ was investigated. But the results were restricted to the case where the associated group of the Hopf algebra H is a safeprime.
Abstract: Let L be a Galois extension of K, fields, with Galois group Γ. We obtain two results. First, if Γ = Hol(Zpe ), we determine the number of Hopf Galois structures on L/K where the associated group of the Hopf algebra H is Γ (i.e. L ⊗K H ∼ L(Γ)). Now let p be a safeprime, that is, p is a prime such that q =( p − 1)/2 > 2 is also prime. If L/K is Galois with group Γ = Hol(Zp), p a safeprime, then for every group G of cardinality p(p−1) there is an H-Hopf Galois structure on L/K where the associated group of H is G, and we count the structures.

18 citations


Journal ArticleDOI
TL;DR: The p-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group G= Gal (k nr,2 /k) for k= Q ( d ) with d=−445,−1015,−1595,−2379 as discussed by the authors.

18 citations


Journal ArticleDOI
Takeshi Torii1
TL;DR: In this article, a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group S n -1 of H n-1.
Abstract: In this note we study a certain formal group law over a complete discrete valuation ring F [[ u n -1 ]] of characteristic p 0 which is of height n over the closed point and of height n -1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law H n -1 of height n - 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group S n -1 of H n -1 . We show that the automorphism group S n of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of S n -1 to the cohomology of S n with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

17 citations


Journal ArticleDOI
01 Nov 2003
TL;DR: In this article, a generic polynomial over the projective line with two parameters was constructed for all transitive subgroups of the symmetric group of degree 5 by considering the action on the moduli space of the projected line with ordered five marked points.
Abstract: In this article, we construct generic polynomials over $\boldsymbol{Q}$ with two parameters for all transitive subgroups of the symmetric group of degree 5 by considering the action on the moduli space of the projective line with ordered five marked points. Although polynomials having such properties are already known, our device is unifying through all the cases, and in some cases we obtain polynomials with much simpler coefficients.

17 citations


Journal ArticleDOI
TL;DR: In this paper, it is shown that the Galois group of an extension L/F puts constraints on the structure of the relative ideal class group Cl(L/F) in the semisimple abelian case.

Journal ArticleDOI
TL;DR: In this article, the authors investigated the problem of second-order linear recurrence over finite fields with prime order and the characteristic polynomial of the relation is irreducible over F. They showed that a finite extension of F and a subgroup M of the multiplicative group of L such that the elements of M may be written, without repetition, so as to form a cyclically closed sequence which obeys the recurrence.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the arithmetic structure of fields F of characteristic p p containing a primitive pth root of unity for which the maximal pro-p Galois group of F is a (finitely generated) Demuskin group.
Abstract: We study the arithmetic structure of fields F of characteristic \( eq p\) containing a primitive pth root of unity for which the maximal pro-p Galois group of F is a (finitely generated) Demuskin group.

Journal ArticleDOI
TL;DR: In this paper, it was shown that R (M) is a subgroup of C l(M ) under the assumption that k and the 3rd cyclotomic field of Q are linearly disjoint and the class number of k is odd.

Journal ArticleDOI
TL;DR: The non-existence of continuous irreducible representations p : Gal(Q/Q) \longrightarrow GL(F) with Artin conductor N outside p for a few small values of p and N is proved in this article.
Abstract: The non-existence is proved of continuous irreducible representations p : Gal(Q/Q) \longrightarrow GL(F) with Artin conductor N outside p for a few small values of p and N.N.

Journal ArticleDOI
TL;DR: For any prime power q, the construction of a polynomial f q (X) ∈ F q (t,u)[X] whose Galois group over F q(t, u) is the Dickson group G 2 (q) has been studied in this paper.
Abstract: For any prime power q we determine a polynomial f q (X) ∈ F q (t,u)[X] whose Galois group over F q (t, u) is the Dickson group G 2 (q). The construction uses a criterion and a method due to Matzat.

Journal ArticleDOI
TL;DR: It is shown that every element in the centre of the Galois group of a given polynomial f ∈ Q[x] along with its action on the set of roots of f, without previously computing the group.
Abstract: We present a new algorithm for computing the centre of the Galois group of a given polynomial f ∈ Q[x] along with its action on the set of roots of f , without previously computing the group. We show that every element in the centre is representable by a family of polynomials in Q[x]. For computing such polynomials, we use quadratic Newton-lifting and truncated expressions of the roots of f over a p-adic number field. As an application we give a method for deciding the nilpotency of the Galois group. If f is irreducible with nilpotent Galois group, an algorithm for computing it is proposed.

Book ChapterDOI
15 Dec 2003
TL;DR: The order of the Galois group of an arbitrary polynomial f(x) ∈ ℤ[x] can be computed in P# P and the order can be approximated by a randomizedPolynomial-time algorithm with access to an NP oracle.
Abstract: Assuming the generalized Riemann hypothesis, we prove the following complexity bounds: The order of the Galois group of an arbitrary polynomial f(x) ∈ ℤ[x] can be computed in P# P. Furthermore, the order can be approximated by a randomized polynomial-time algorithm with access to an NP oracle. For polynomials f with solvable Galois group we show that the order can be computed exactly by a randomized polynomial-time algorithm with access to an NP oracle. For all polynomials f with abelian Galois group we show that a generator set for the Galois group can be computed in randomized polynomial time.

Journal Article
TL;DR: A new approach is given based on the idea of using resultant in order to know the common factors between an empirical polynomial and a generic polynomials in its neighborhood.
Abstract: In this paper the concept of neighborhood of a polynomial is analyzed. This concept is spreading into Scientific Computation where data are often uncertain, thus they have a limited accuracy. In this context we give a new approach based on the idea of using resultant in order to know the common factors between an empirical polynomial and a generic polynomial in its neighborhood. Moreover given a polynomial, the Square Free property for the polynomials in its neighborhood is investigated.

Journal Article
TL;DR: In this article, it was shown that the order of the Galois group of an arbitrary polynomial f(x) ∈ ℤ[x] can be computed in P#P with access to an NP oracle.
Abstract: Assuming the generalized Riemann hypothesis, we prove the following complexity bounds: The order of the Galois group of an arbitrary polynomial f(x) ∈ ℤ[x] can be computed in P# P Furthermore, the order can be approximated by a randomized polynomial-time algorithm with access to an NP oracle For polynomials f with solvable Galois group we show that the order can be computed exactly by a randomized polynomial-time algorithm with access to an NP oracle For all polynomials f with abelian Galois group we show that a generator set for the Galois group can be computed in randomized polynomial time

Patent
21 Feb 2003
TL;DR: A method and apparatus for performing Galois field multiplication with reduced redundancy is described in this paper. But the method is not suitable for the case of binary polynomials, and it requires the use of a premultiplier logic component.
Abstract: A method and apparatus for performing Galois field multiplication with reduced redundancy. Generally, multiplication by a Galois field multiplier involves the multiplication of two polynomials modulo another polynomial. The Galois field multiplier has two Galois Field elements in a field of GF(2 n ) that correspond to the binary polynomials A[X] and B[X]: A[X]=a n-1 X n-1 +a n-2 X n-2 +a n-3 X n-3 + . . . a 1 X+a 0 , B[X]=b n-1 X n-1 +b n-2 X n-2 +b n-3 X n-3 + . . . b 1 X+b 0 , where n corresponds to a number of terms in a Galois extension field of the Galois multiplier, and n-1 is an order of the polynomial A[X]. Premultiplier logic translates the binary polynomial A[X] into a binary vector c r , where r is the number of terms of the vector. The premultiplier logic is configured to modulo-2 add together various coefficients of the a 0 through a n-1 , coefficients to produce various terms c 0 through c r-1 of the c r binary vector. Binary multiplication and addition logic then operates on the c 0 through c r-1 coefficients and the b 0 through b n-1 coefficients to produce d 0 through d n coefficients of a binary polynomial D[X]. The coefficients d 0 through d n are the output of the Galois field multiplier. Utilization of the premultiplier logic component reduces the amount of binary multiplication and addition logic needed to produce the coefficients d 0 through d n of the binary polynomial D[X].

Journal ArticleDOI
TL;DR: In this article, it was shown that a split group extension G' of Sp by a p-elementary group, a G-faithful quasi-permutation ZG'-lattice M, and a one-cocycle α in Ext 1 G'(M, F*) such that Cp is stably isomorphic to F α (M) G'.
Abstract: Let G be a finite group, let M be a ZG-lattice, and let F be a field of characteristic zero containing primitive p th roots of 1. Let F(M) be the quotient field of the group algebra of the abelian group M. It is well known that if M is quasi-permutation and G-faithful, then F(M) G is stably equivalent to F(ZG) G . Let C n be the center of the division ring of n × n generic matrices over F. Let S n be the symmetric group on n symbols. Let p be a prime. We show that there exist a split group extension G' of Sp by a p-elementary group, a G'-faithful quasi-permutation ZG'-lattice M, and a one-cocycle α in Ext 1 G'(M, F*) such that Cp is stably isomorphic to F α (M) G' . This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if F is algebraically closed, there is a group extension E of Sp by an abelian p-group such that Cp is stably equivalent to the invariants of the Noether setting F(E).

Journal ArticleDOI
TL;DR: In this paper, the splitting field of an irreducible polynomial of odd degree n > 1 whose Galois group is a Frobenius group was shown to have a tower of nor subfields.
Abstract: Let f (x) be an irreducible polynomial of odd degree n > 1 whose Galois group is a Frobenius group. We suppose that the Frobenius complement is a cyclic group of even order h. Let 2 t h. For each i ¼ 1, 2, ... , t we show that the splitting field L of f (x) has exactly one subfield Ki with [Ki : Q] ¼ 2 i . These subfields form a tower of nor

Journal ArticleDOI
TL;DR: For a cyclic Galois extension M = L(α 1/n ) of L of degree n such that M is Galois over F, this article gave an explicit parametrization of those a that lead to each possible group Gal(M/F).
Abstract: Let n be any integer with n > 1, and let F C L be fields such that [L: F] = 2, L is Galois over F, and L contains a primitive n th root of unity ζ. For a cyclic Galois extension M = L(α 1/n ) of L of degree n such that M is Galois over F, we determine, in terms of the action of Gal(L/F) on a and ζ, what group occurs as Gal(M/F). The general case reduces to that where n = p e , with p prime. For n = p e , we give an explicit parametrization of those a that lead to each possible group Gal(M/F).

Journal ArticleDOI
TL;DR: In this article, the 0,1 distribution in the highest level sequence of a primitive sequence over Z 2 e generated by a primitive polynomial of degree n has been studied, and it is shown that the larger n is, the closer to 1/2 the proportion of 1 will be.
Abstract: In this paper, we discuss the 0,1 distribution in the highest level sequence a e -1 of primitive sequence over Z 2 e generated by a primitive polynomial of degree n . First we get an estimate of the 0,1 distribution by using the estimates of exponential sums over Galois rings, which is tight for e relatively small to n . We also get an estimate which is suitable for e relatively large to n . Combining the two bounds, we obtain an estimate depending only on n , which shows that the larger n is, the closer to 1/2 the proportion of 1 will be.

Posted Content
TL;DR: This work considers the Sylvester resultant of f and g, where f is a generic polynomial of degree 2 or 3 and g is ageneric polynometric of degree n, and finds tight asymptotics for the resultant's height.
Abstract: Let n be a positive integer. We consider the Sylvester Resultant of f and g, where f is a generic polynomial of degree 2 or 3 and g is a generic polynomial of degree n. If f is a quadratic polynomial, we find the resultant's height. If f is a cubic polynomial, we find tight asymptotics for the resultant's height.

Journal Article
TL;DR: In this paper, the use of a principal ideal domain structure for the analysis and design of multidimensional systems is discussed, where a lattice structure can be introduced for IO-relations generated by polynomial matrices in a signal space X (an Abelian group).
Abstract: The use of a principal ideal domain structure for the analysis and design of multidimensional systems is discussed. As a first step it is shown that a lattice structure can be introduced for IO-relations generated by polynomial matrices in a signal space X (an Abelian group). It is assumed that the matrices take values in a polynomial ring F[p] where F is a field such that F[p] is a commutative subring of the ring of endomorphisms of X. After that it is analysed when a given F[p] acting on X can be extended to its field of fractions F(p). The conditions on the pair (F[p],X) are quite restrictive, i.e. each non-zero a(p) 2 F[p] has to be an automorphism on X before the extension is possible. However, when this condition is met, say for operators {p1,p2,...,pn 1}, a polynomial ring F[p1,p2,...,pn] acting on X can be extended to F(p1,p2,...,pn 1)[pn], resulting in a principal ideal domain structure. Hence in this case all the rigorous principles of ‘ordinary’ polynomial systems theory for the analysis and design of systems is applicable. As an example, both an observer for estimating non-measurable outputs and a stabilizing controller for a distributed parameter system are designed.

Journal ArticleDOI
TL;DR: In this paper, a Galois algebra with Galois group G is characterized and the Galois extension B e, generated by a Boolean algebra generated by G with a central idempotent element in B a, and H e = { g ∈ G | e e e g = e }.
Abstract: Let B be a Galois algebra with Galois group G , J g = { b ∈ B | b x = g ( x ) b for all x ∈ B } for each g ∈ G , and B J g = B e g for a central idempotent e g , B a the Boolean algebra generated by { 0 , e g | g ∈ G } , e a nonzero element in B a , and H e = { g ∈ G | e e g = e } . Then, a monomial e is characterized, and the Galois extension B e , generated by e with Galois group H e , is investigated.

Journal ArticleDOI
TL;DR: For a given field F of characteristic 0, the authors considers a normal extension E/F of finite degree d and finite Abelian subgroups G⊂GLn(E) of a given exponent t and shows that any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.
Abstract: For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups G⊂GLn(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices g∈G to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.

01 Jan 2003
TL;DR: In this article, the Galois group of an irreducible quintic polynomial 2Z[x] was shown to not always belong to the 2Z group.
Abstract: In [4, Proposition, pp. 883{884] a procedure is given to nd the Galois group of an irreducible quintic polynomial 2Z[x]. It is shown that this procedure does not always nd the Galois group.