## Two Step Descent In Modular Galois Theory, Theorems Of Burnside And Cayley, And Hilbert's Thirteenth Problem

### BibTeX

@MISC{Abhyankar_twostep,

author = {Shreeram S. Abhyankar},

title = {Two Step Descent In Modular Galois Theory, Theorems Of Burnside And Cayley, And Hilbert's Thirteenth Problem},

year = {}

}

### OpenURL

### Abstract

We propound a Descent Principle by which previously constructed equations over GF(q n )(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups, where q = p u ? 1 is a power of a prime p and n is a positive integer. Currently this is achieved by starting with a vectorial (= additive) q-polynomial of q-degree m with Galois group GL(m; q) where m is any positive integer and then, under suitable conditions, enlarging its Galois group to GL(m; q n ) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degree n. So, alternatively, we may regard this as an Ascent Principle. Elsewhere we proved this when m is square-free with GCD(mnu; 2p) = 1 = GCD(m;n). There the proof was based on CT (= the Classification Theorem of Finite Simple Groups) in its incarnation of CPT (= the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors). Here, without ...