Galois Groups of Generalized Iterates of Generic Vectorial Polynomials

✍ Scribed by Shreeram S. Abhyankar; Ganapathy S. Sundaram

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 175 KB
• Volume: 7
• Category: Article
• ISSN: 1071-5797
• DOI: 10.1006/ffta.2000.0302

✦ Synopsis

Let q"pS'1 be a power of a prime p, and let k O be an over"eld of GF(q). Let m'0 be an integer, let J* be a subset of +1, 2 , m,, and let E* KO (>)"> qK # HZ( * X H >O K\H where the X H are indeterminates. Let J ? be the set of all m! where is either 0 or a divisor of m di!erent from m. Let s(¹)" 04i4n s G ¹G be an irreducible polynomial of degree n'0 in ¹ with coe$cients s G in GF (q). Let

, where E* G KO (>), is the ordinary ith iterate. We prove that if J ? LJ*, m is square-free, and GCD (m, n)"1"GCD (mnu, 2p), then Gal (E* Q KO , k O (+X H : j3J*,)"G¸(m, qL). The proof is based on CT ("the Classi"cation Theorem of Finite Simple Groups) in its incarnation as CPT ("the Classi"cation of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors). 2000 Academic Press 1. INTRODUCTION Throughout this paper let q"pS'1 be a power of a prime p, let m'0 and n'0 be integers, and let GF(q)Lk O LKL be "elds where is an