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Showing papers on "Generic polynomial published in 1980"


Journal ArticleDOI

32 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the trinomial Xn + ΣΓ^o1*^* has galois group Sn over F(T, u) under mild conditions involving p(>0) and that the results are always valid if F has characteristic zero and hold under mild condition involving the characteristic of F otherwise.
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

24 citations



Journal ArticleDOI
David J. Saltman1
TL;DR: A Grunwald-Wang-type theorem whenever there is a generic extension for a group G over a field F and such that L superset, dbl equals F is found.
Abstract: We define the notion of a generic Galois extension with group G over a field F. Let R be a communtative ring of the form F[x1,..., xn](1/s) and let S be a Galois extension of R with group G. Then S/R is generic for G over F if the following holds. Assume K/L is a Galois extension of fields with group G and such that L ⊇ F. Then there is an F algebra map f:R → L such that K ≅ S [unk]RL. We construct generic Galois extensions for certain G and F. We show such extensions are related to Noether's problem and the Grunwald-Wang theorem. One consequence is a simple proof of known counter examples to Noether's problem. On the other hand, we have an elementary proof of a chunk of the Grunwald-Wang theorem, and in a more general context. In fact, we have a Grunwald-Wang-type theorem whenever there is a generic extension for a group G over a field F.

11 citations