Name
Abstract | ||
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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 ⊆ L and Gal ( N / L ) ≤ G arises as the splitting field of a suitable specialization g ( λ 1 , . . . , λ m , 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 ). References References 1 S.S. Abhyankar Galois embeddings for linear groups Trans. Am. Math. Soc. (to appear.) 2000 2 E.V. Black Deformations of dihedral 2-group extensions of fields Trans. Am. Math. Soc. 351 1999 3229 3241 3 J. Buhler Z. Reichstein On the essential dimension of a finite group Compos. Math. 106 1997 159 179 4 D. Carlisle P.H. Kropholler Rational invariants of certain orthogonal and unitary groups Bull. London Math. Soc. 24 1992 57 60 5 F.R. DeMeyer Generic polynomials J. Algebra 84 1983 441 448 6 R. Hartshorne Algebraic Geometry 1977 Springer-Verlag New York, Heidelberg, Berlin 7 N. Jacobson Basic Algebra, volume 1 1985 Freeman New York 8 G. Kemper A constructive approach to Noether’s problem Manuscripta Math. 90 1996 343 363 9 G. Kemper G. Malle The finite irreducible linear groups with polynomial ring of invariants Transformation Groups 2 1997 57 89 10 G. Kemper G. Malle Invariant fields of finite irreducible reflection groups Math. Ann. 315 1999 569 586 11 W. Kuyk On a theorem of E. Noether Nederl. Akad. Wetensch. Proc. Ser. A 67 1964 32 39 12 O. Lecacheux Constructions de polynomes generiques a groupe de Galois resoluble Acta Arith. 86 1998 207 216 13 A. Ledet Generic and explicit realisation of small p -groups J. Symb. Comput. 30 2000 859 865. doi:10.1005/jsco.2000.0386 14 A. Ledet Generic extensions and generic polynomials J. Symb. Comput. 30 2000 867 872. doi:10.1005/jsco.2000.0387 15 G. Malle B.H. Matzat Inverse Galois Theory 1999 Springer-Verlag Berlin, Heidelberg 16 T. Miyata Invariants of certain groups I Nagoya Math. J. 41 1971 69 73 17 H. Nakajima Invariants of finite groups generated by pseudo-reflections in positive characteristic Tsukuba J. Math. 3 1979 109 122 18 E. Noether Gleichungen mit vorgeschriebener gruppe Math. Ann. 78 1918 221 229 19 D.J. Saltman Generic Galois extensions and problems in field theory Adv. Math. 43 1982 250 283 20 D.J. Saltman Retract rational fields and cyclic Galois extensions Isr. J. Math 47 1984 165 215 21 F. Seidelmann Die Gesamtheit der kubischen und biquadratischen Gleichungen mit Affekt bei beliebigem Rationalitätsbereich Math. Ann. 78 1918 230 233 22 G.C. Shephard J.A. Todd Finite unitary reflection groups Can. J. Math. 6 1954 274 304 23 G.W. Smith Generic cyclic polynomials of odd degree Commun. Algebra 19 1991 3367 3391 24 L. Smith Polynomial Invariants of Finite Groups 1995 A. K. Peters Wellesley, Mass 25 A. Speiser Zahlentheoretische Sätze aus der Gruppentheorie Math. Z. 5 1919 1 6 26 C. Wilkerson A primer on the Dickson invariants Am. Math. Soc. Contemp. Math. Series 19 1983 421 434 |
Year | DOI | Venue |
|---|---|---|
2000 | 10.1006/jsco.1999.0385 | J. Symb. Comput. |
Keywords | DocType | Volume |
generic polynomial | Journal | 30 |
Issue | ISSN | Citations |
6 | Journal of Symbolic Computation | 1 |
PageRank | References | Authors |
0.38 | 2 | 2 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Gregor Kemper | 1 | 70 | 11.53 |
| Elena Mattig | 2 | 1 | 0.38 |