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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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28 citations
TL;DR: In this article, the Galois group of an integral polynomial is computed by resolvent computation by modular techniques, based on an exact method to find integral roots of relative resolvents by direct evaluation of invariants over some padic number field or its extension.
28 citations
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
01 Sep 1996
TL;DR: In this article, the solvability of polynomials with primitive Galois groups was shown to be solvable in polynomial time, by using a primitive group on its roots.
Abstract: This study is a continuation of Yokoyama et al. [22], which improved the method by Landau and Miller [11] for the determination of solvability of a polynomial over the integers. In both methods, the solvability of a polynomial is reduced, in polynomial time, to that of polynomials, each of which is constructed so that its Galois group acts primitively on its roots. Then, by virtue of Palfy’s bound [14], solvability of polynomials with primitive Galois groups can be determined in polynomial time. An effective method, thus, exists in theory. For practical computation, however, the most serious problem remains: How to determine solvability of each polynomial with primitive Galois group.
27 citations
TL;DR: In this paper, the authors find a relationship between regular embeddings of G, an elementary abelian p-group of order p n, into unipotent upper triangular matrices with entries in F p and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices.
26 citations