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Generic Polynomials with Few Parameters

✍ Scribed by Gregor Kemper; Elena Mattig

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 335 KB
Volume: 30
Category: Article
ISSN: 0747-7171
DOI: 10.1006/jsco.1999.0385

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✦ Synopsis

We call a polynomial g(t 1 , . . . , tm, 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 ⊆ L and Gal(N/L) ≤ G arises as the splitting field of a suitable specialization g(λ 1 , . . . , λm, X) with λ i ∈ 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).

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