Generic polynomials with few parameters

被引:20
作者
Kemper, G
Mattig, E
机构
[1] Univ Heidelberg, IWR, D-69120 Heidelberg, Germany
[2] Univ Essen Gesamthsch, Fachbereich Math 6 & Informat, D-45117 Essen 1, Germany
关键词
D O I
10.1006/jsco.1999.0385
中图分类号
TP301 [理论、方法];
学科分类号
081202 ;
摘要
We call a polynomial g(t(1),...,t(m),X) over a field K generic for a group G if it has Galois group G as a polynomial in X, and if every Galois field extension N/L with K subset of or equal to L and Gal(N/L) less than or equal to G arises as the splitting field of a suitable specialization g(lambda (1),...,lambda (m),X) with lambda (i) is an element of L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic polynomial which is often particularly simple and therefore useful. Then we consider various examples and applications in characteristic 0 and in positive characteristic. These include results on so-called vectorial polynomials and a generalization of an embedding criterion given by Abhyankar. We give recursive formulas for generic polynomials over a field of defining characteristic for the groups of upper unipotent and upper triangular matrices, and explicit formulae for generic polynomials for the groups GU(2)(q(2)) and GO(3)(q). (C) 2000 Academic Press.
引用
收藏
页码:843 / 857
页数:15
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