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Showing papers on "Generic polynomial published in 1995"


Journal ArticleDOI
TL;DR: For an integer n ≥ 14, this paper showed that the splitting field of the polynomial ǫ(x ) can be embedded in a field with absolute Galois group isomorphic to A n, the double cover of A n, if and only if n ≡ 1 (mod 8).

30 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that it is always possible to transform the problem of tinding the mots of a generic polynomial to determining the eigenvalues of tridiagonal matrices having only simple eigen values.

20 citations


Book ChapterDOI
TL;DR: A systematic and formal method to compute the Galois group of a non-necessarily irreducible polynomial, based on a formal method of specialization of relative resolvents, which reduces the problem to that of specializing a primitive element.
Abstract: We propound a systematic and formal method to compute the Galois group of a non-necessarily irreducible polynomial: we proceed by successive inclusions, using mostly computations on scalars (and very few on polynomials). It is based on a formal method of specialization of relative resolvents: it consists in expressing the generic coefficients of the resolvent using the powers of a primitive element, thanks to a quadratic space structure; this reduces the problem to that of specializing a primitive element, which we are able to do in the case of the descending by successive inclusions. We incidentally supply a way to make separable a resolvent.

18 citations


Journal ArticleDOI
TL;DR: In this article, the Galois group of polynomial relations between x1,..., xn may or may not exist, resp. which relations cannot exist and the conditions on which relations may or cannot exist.

17 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that if (H, H1) is a pseudogroup generated by a finite numberH1 of germs of conformal diffeomorphisms of ℂ defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H1)={g′(0)/g∈(H,H 1)}⊂ℂ* is dense in D, then the set of fixed points of H, H 1 is dense.
Abstract: We show in this paper Theorem 2 that if (H, H1) is a pseudogroup generated by a finite numberH1 of germs of conformal diffeomorphisms of ℂ defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H1)={g′(0)/g∈(H,H1)}⊂ℂ* is dense in ℂ*, then the set of fixed points of the pseudogroup (H, H1) is dense inD. This implies the abundance of distinct homotopy classes of loops in leaves of foliations defined in ℂ2 by generic polynomial vector fields as well as for germs of holomorphic vector fields in ℂ2 beginning with generic jets, both of degree at least 2. These homotopy classes may be realized arbitrarily close to the line at infinity or to 0, respectively. This shows the genericity of polynomial vector fields with infinite Petrovsky-Landis genus ([5]).

9 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that certain polynomials of degree ≥ 3 have a cyclic group of order n as Galois group over all rational function fields with characteristic not dividing n.
Abstract: It is proved that certain polynomials of degree ≱3 have a cyclic group of order n as Galois group over all rational function fields k(t) with characteristic not dividing n. Moreover, the extension fields of k(t) generated by the polynomials have k as precise field of constants, and possess an unramified rational point. For all 3≤≰20 with the exceptions of 17 and 19 the polynomials are calculated explicitly

9 citations


Journal ArticleDOI
TL;DR: In this article, a commutative diagram of the symmetric group 2−Sn is given, where 2−sn is the double cover of symmetric groups and Sn is the non-trivial double cover An of the alternating group An in which transpositions lift to elements of order 4 and the morphism j− is injective.
Abstract: where 2−Sn is the double cover of the symmetric group Sn reducing to the non-trivial double cover An of the alternating group An in which transpositions lift to elements of order 4 and the morphism j− is injective. We identify 2−Sn with j−(2−Sn) and note that if {xs}s∈Sn is a system of representatives of Sn in 2−Sn, we can take it as a system of representatives of Sn in 2Sn and so 2Sn is determined modulo isomorphisms. If c denotes a generator of C2r , the elements of 2Sn can be written as cxs, for s ∈ Sn, 0 ≤ i ≤ 2 − 1. We note that H := {cxs : s ∈ An, i = 0, 2r−1} ∪ {cxs : s ∈ Sn \ An, i = 2r−2, 3 · 2r−2} is a subgroup of 2Sn, isomorphic to 2Sn, the second double cover of the symmetric group Sn reducing to An. We then obtain a commutative diagram 2Sn Sn

4 citations


Book ChapterDOI
22 Aug 1995
TL;DR: For polynomials over the integers or rationals, it is known that this problem is exponential space complete as mentioned in this paper, and the complexity results known for a number of problems related to polynomial ideals are discussed.
Abstract: A polynomial ideal membership problem is a (w+1)-tuple P=(f, g1,g2, ..., g w ) where f and the g i are multivariate polynomials over some ring, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, it is known that this problem is exponential space complete. We discuss complexity results known for a number of problems related to polynomial ideals, like the word problem for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz, and the reachability and other problems for (reversible) Petri nets.

2 citations


Journal Article
TL;DR: In this paper, an injective envelope description of an arbitray simple module over a polynomial ring over a field K in indeterminates is given, where the envelopes are defined in terms of a set of parameters.
Abstract: In [9], we gave a very explicit description of the injective envelope of an arbitray simple module over a polynomial ring $K[X_1, \ldots, X_n]$ over a field K in indeterminates $X_1, \ldots, X_n$. This paper presents another approach to give a description.

2 citations


Journal ArticleDOI
01 Aug 1995
TL;DR: In this paper, the authors prove that vectors from these two bases are as independent as possible, subject to a hypothesis on the Galois group, and show that the two bases for rational numbers are independent.
Abstract: A givenn ×n matrix of rational numbers acts onCπ and onQπ. We assume that its characteristic polynomial is irreducible and compare a basis of eigenvectors forCπ with the standard basis forQπ. Subject to a hypothesis on the Galois group we prove that vectors from these two bases are as independent of each other as possible.

Journal ArticleDOI
TL;DR: In this article, the necessary and sufficient conditions on the field F in order to have a Galois and cogalois extension K F with Galois group G were given. But these conditions were only applicable to a number field F and an arbitrary allowable group G.

Journal ArticleDOI
01 Jan 1995
TL;DR: In this paper, the fraction field of the Iwasawa algebra Λ = Zp[[T ]] is topologized so that a base of neighborhoods of 0 is given by powers of M, and define neighborhoods of other elements of Λ by translation.
Abstract: We find an appropriate topology to put on K, the fraction field of the Iwasawa algebra Λ = Zp[[T ]], so that compact subgroups of K are in fact contained in Λ. This ensures that Galois representations to K have image in Λ. Let Λ = Zp[[T ]] be the Iwasawa algebra. Λ is a unique factorization domain. The p-adic Weierstrass Preparation Theorem says that elements of Λ may be represented as uf , where f is a polynomial and u is a unit. Let M = (p, T ) be the maximal ideal of Λ. Topologize Λ so that a base of neighborhoods of 0 is given by powers of M , and define neighborhoods of other elements of Λ by translation. Let K be the field of fractions of Λ. The first question to consider is how to topologize K. One somewhat obvious approach is to say that a set U ⊆ K is open in K precisely when kU ∩Λ is an open subset of Λ for all k ∈ K. This definition makes addition and multiplication continuous. Topologized in this way, a compact subset of GLn(K) which is also a subgroup is conjugate to a subset of GLn(Λ). Unfortunately, there is one major drawback to this topology. Proposition. The function f(x) = x is not continuous in this topology. Proof. There are many ways to see this. Perhaps the simplest is to observe that the sequence an = p+ T n converges to p. However, the sequence a n is closed, since for a fixed k ∈ K , ka n will be an element of Λ for only finitely many n. Hence, a n cannot converge to p . We therefore need a different topology on K, and fortunately there is an obvious candidate. If λ ∈ Λ, we can define v(λ) = n if λ ∈ M and λ 6∈ M and v(0) = ∞. Krull’s Theorem [1] implies that ⋂ M = {0}, and so the function v is well-defined. Lemma. v is a valuation on Λ. Proof. Let f, g ∈ Λ. Set v(f) = m and v(g) = n. Obviously, v(f + g) ≥ min(v(f), v(g)), so we need only show that v(fg) = v(f) + v(g). Use the Weierstrass Preparation Theorem to write f = u1f , g = u2g , where u1 and u2 are units and f ′ and g are polynomials. Write f ′ = ∑ aiT i and g = ∑ bjT . Let vp be the usual p-adic valuation on Zp. Of those terms in ∑ aiT i with v(aiT ) = m, let akT k be the term so that vp(ak) is minimal. (It is easy to see that there is a unique minimum, because if v(aiT ) = m, then vp(ai) = m− i.) Similarly, let blT l be the term in the second sum minimizing vp(bl) subject to v(blT ) = n. If we now consider the coefficient ck+l of T k+l in the product fg = u1u2f g, we see that vp(ck+l) = vp(ak) + vp(bl). Therefore v(ck+lT ) = m+ n, and we finally have v(fg) = m+ n. This lemma in fact is true in considerably greater generality, but the statement does not seem to appear in the literature in this form. 1991 Mathematics Subject Classification. 22C05, 11S20.