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Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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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.
1 citations
15 Aug 2005
TL;DR: In this paper, the authors give explicitly the whole family of Galois extensions of K with Galois group 2S 4 * Q 8 or 2S4 * D 8 and determine the discriminant of such an extension.
Abstract: For K a field containing the finite field F g we give explicitly the whole family of Galois extensions of K with Galois group 2S 4 * Q 8 or 2S 4 * D 8 and determine the discriminant of such an extension.
1 citations
TL;DR: In this paper, it was shown that for almost all odd prime powers p m where c divides p m − 1, if f is a polynomial over GR (p n, m) satisfying certain conditions, then there is a primitive element γ in the ring such that f (γ) is a c th power.
1 citations
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TL;DR: For a system of indeterminates (X_a) indiced by the projective plane over F_2, there exists a 3-3 correspondance compatible with the incidence structures of P^2(2), such that (Xa) is one of the orbits of it as discussed by the authors.
Abstract: We prove that for a system of indeterminates (X_a) indiced by the P^2(2), the projective plane over F_2, there exists a 3-3 correspondance compatible with the incidence structures of P^2(2), such that (X_a) is one of the orbits of it.
We give two applications of this construction :
1) for any sufficientely general polynomial P in k[X] over a field k of car. 0, such that its Galois group is a subgroup of L3(2) ((=L2(7)), there exists Q in k[X] such that the Galois group of P-TQ over k(T) is L3(2). This implies in particular the so-called "arithmetical lifting property" for L3(2) over k.
2) There exists a generic polynomial in 7 parameters for polynomials of degree 7 with Galois group L3(2). This is equivalent to the fact that the Noether's problem for L3(2) acting over the seven points of P^2(2) has a positive answer.
1 citations
TL;DR: In this paper, it was shown that there is a finite polynomial extension F : k n → k n such that gdeg F ≤ ( gdeg f ) k + 1.
1 citations