Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.

The Galois group of a polynomial with two indeterminate coefficients

Abstract: Φ 0) is a polynomial in which two of the coefficients are indeterminates t, u and the remainder belong to a field F. We find the galois group of / over F(t, u). In particular, it is the full symmetric group Sn provided that (as is obviously necessary) /(X) Φ fχ(Xr) for any r > 1. The results are always valid if F has characteristic zero and hold under mild conditions involving the characteristic of F otherwise. Work of Uchida [10] and Smith [9] is extended even in the case of trinomials Xn + tXa + u on which they concentrated. 1* Introduction* Let F be any field and suppose that it has characteristic p, where p — 0 or is a prime. In [9], J. H. Smith, extending work of K. Uchida [10], proved that, if n and a are coprime positive integers with n > α, then the trinomial Xn + tXa + u, where t and u are independent indeterminates, has galois group Sn over F(t, u), a proviso being that, if p > 0, then p \ na(n — α). (Note, however, that this conveys no information whenever p — 2, for example.) Smith also conjectured that, subject to appropriate restriction involving the characteristic, the following holds. Let I be a subset (including 0) of the set {0, 1, , n — 1} having cardinality at least 2 and such that the members of / together with n are co-prime. Let T — {ti9 i e 1} be a set of indeterminate s. Then the polynomial Xn + ΣΓ^o1*^* has galois group Sn over F(T). In this paper, we shall confirm this conjecture under mild conditions involving p(>0), thereby extending even the range of validity of the trinomial theorem. In fact, we also relax the other assumptions. Specifically, we allow some of the tt to be fixed nonzero members of F and insist only that two members of T be indeterminates. Indeed, even if the co-prime condition is dispensed with, so that the galois group is definitely not SΛ, we can still describe

Polynomial Time Algorithms for Galois Groups

Abstract: In this paper we present several polynomial time algorithms for Galois groups. We show: (i) There are polynomial time algorithms to determine: (a) If the Galois group of an irreducible polynomial over Q is a p-group. (b) the prime divisors of the order of a solvable Galois group (ii) Using the classification theorem for finite simple groups, there is a polynomial time algorithm to determine whether an irreducible polynomial over Q has Galois group Sn or An.

Endomorphisms of superelliptic jacobians

Abstract: Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, \({\mathbb{Z}[\zeta_p]}\) the ring of integers in the pth cyclotomic field, Cf, p : yp = f(x) the corresponding superelliptic curve and J(Cf, p) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(Cf, p) coincides with \({\mathbb{Z}[\zeta_p]}\) . The same is true if n = 4, the Galois group of f(x) is the full symmetric group S4 and K contains a primitive pth root of unity.