Extension of Constants, Rigidity and
Two Conjectures Transparent to the B(ranch) C(ycle) L(emma)

There is an exposition on the BCL in deflist-cov/Branch-Cycle-Lem.html. The 1st point of this paper was to take the most practical examples of problems that had been around for years, and to show how immediately the BCL can produce counterexamples to them.
The group theory is elementary, requiring just the differentiation of some conjugacy classes (from their cycle type) in A_n, the alternating group of degree n.

The 2nd point was to show how polynomial covers f: x f(x)=z, provide an elmentary yet, distinctly nontrivial setting to practice using  families of covers. The author had seen these conjectures announced for several years, explained their falseness to various lecturers and authors, and had yet found there was no response. Thus, this paper. A fuller – and deeper – gamut of elementary BCL uses appear in paplist-mt/fried-kop97.html, paplist-cov/GQpresentation.html and paplist-cov/ser_gal.html their attached pdf files.


We assume understood  the monodromy group, Ĝ_f,K of f over a field K: The Galois group of the splitting field of f(x)=z over K(z), with z a transcendental. Of course, we assume f has coefficients in K, the case K=Q being the territory of the stated conjectures. When K is the algebraic closure of Q, then we drop the hat, and use just the notation G_f. This is the geometric monodromy of f. Recall that the cyclic polynomial of degree n is xⁿ, and the nth Chebychev polynomial is – for an algebraist – the famed polynomial map from calculus cos(ϑ) → cos(nϑ).

The Three Problems: To remove trivial counterexamples known to the proposers of these problems we have two assumptions:

1. f does not decompose as two positive degree polynomials of lower degree (it is indecomposable); and
2. f is neither, by affine change of x or z, a cyclic nor a Chebychev polynomial.

Note: As was proved in an elementary, albeit essential, piece of the resolution of the Schur Conjecture (say, [§4.b, paplist-ff/sch-carlitz.pdf]), indecomposability of f (just for polynomials, not rational functions) is independent of the choice of its definition field. Here is the statement of the three conjectures, under assumptions #1 and #2.

3. Cohen: There is no extension of constants (that is, Ĝ _f,K=G_f).
4. Chowla-Zassenhaus: For a prime p large, f(x)-b does not split into linear factors for any b ∈Z/p.

The Upshot of the Paper: Problems #3 and #4 are equivalent, and for each odd non-square integer n, there is a polynomial counterexample to each conjecture. The counterexamples are polynomials we call (A_n,S_n)-realizations: Geometric monodromy group is A_n, but over Q the monodromy (arithmetic) is S_n. Further, using Muller's classification of the monodromy groups of polynomials, we show n must be odd.

Later, Muller completed the result by showing that our conditions were exact, nonsquare is necessary. The paper also needed to eliminate polynomials in the Davenport families of polynomials  that arose in such considerations as the genus 0 problem.

Finding all actual Nielsen classes that can possibly produce (A_n,S_n)-realizations is the toughest problem left by the paper. Our examples tie nicely between these elementary results and considerably harder problems that require some version of Hurwitz spaces to complete the analysis.