Abhyankar, Shreeram S. ; Sundaram, Ganapathy S.
(2001)
*Galois groups of generalized iterates of generic vectorial polynomials*
Finite Fields and Their Applications, 7
(1).
pp. 92-109.
ISSN 1071-5797

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1006/ffta.2000.0302

## Abstract

Let q=p^{u}>1 be a power of a prime p, and let k_{q} be an overfield of GF(q). Let m>0 be an integer, let J^{∗} be a subset of {1,...,m}, and let E^{∗} _{m}, _{q}(Y)=Y^{qm}+∑ _{j ∈j∗}X _{j}Y^{qm-j }where the X_{j }are indeterminates. Let J^{+} be the set of all m− v where v is either 0 or a divisor of m different from m. Let s(T)=∑ 0 ≤ i ≤ n s_{i}T^{i} be an irreducible polynomial of degree n>0 in T with coefficients s_{i} in GF(q). Let E^{*[s]}, _{m,q}(Y) be the generalized sth iterate of E^{*}_{m,q}(Y); i.e., E^{*}^{[s]}_{m,q}(Y )=∑ 0 ≤ i ≤ n s_{i} E^{*[i]}_{m,q} (Y), where E^{*[i] }_{m,q} (Y), is the ordinary ith iterate. We prove that if J^{+} ⊂,j^{*}, m is square-free, and GCD(m,n)=1=GCD(mnu,2p), then Gal(E^{[s]}_{m,q}, k_{q}({X _{j}:∈ j^{*}})=GL(m, q^{n}). The proof is based on CT (=the Classification Theorem of Finite Simple Groups) in its incarnation as CPT (=the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors).

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Deposited On: | 16 Sep 2010 06:55 |

Last Modified: | 04 Jul 2011 07:06 |

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