Michael E. Zieve:

**Bivariate factorizations via Galois theory, with
application to exceptional polynomials**,

*J. Algebra*
**210** (1998), 670–689.
**MR** 99m:12001
(The
published
version is available online.)

We give a general method for factoring
*R*_{g} := *g*(*X*) - *g*(*Y*),
where *g* is a polynomial over a field *K*.
Our approach often works when *g* varies over an infinite family of polynomials.

Factorizations of *R*_{g} are especially important when *g* is
an *exceptional polynomial*, which by definition means that the scalar multiples of
*X* - *Y* are the only absolutely irreducible factors of
*R*_{g} in *K*[*X*, *Y*].
Exceptional polynomials arise in various investigations in case *K* is finite,
since in that case a polynomial *g* is exceptional if and only if the map
*a* → *g*(*a*) induces a bijection on infinitely
many finite extensions of *K*. We apply our method to factor
*R*_{g} for each *g* in the infinite
family of exceptional polynomials discovered recently by
Lenstra and Zieve. We also apply our method to factor
*R*_{g} in case *g* is one of the
Müller–Cohen–Matthews exceptional polynomials; in this
case these factorizations had been discovered previously by more complicated methods.

*Additional comment from September 2008*:
The results of this paper were used in the subsequent paper by
Guralnick and Zieve which, together with a paper by
Guralnick, Rosenberg and Zieve, yields a complete
classification of indecomposable exceptional polynomials of non-prime power degree.

*Michael Zieve*:
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