×

A new family of exceptional polynomials in characteristic two. (English) Zbl 1214.12002

Let \(k\) be a field of characteristic \(p\geq 0\), let \(f(X)\in k[X]\setminus k\) and let \(k\) be an algebraic closure of \(k\). A polynomial in \(k[X,Y]\) is called “absolutely irreducible” if it is irreducible in \(k[X,Y]\); and \(f\) is said to be “exceptional” if \(f(X)- f(Y)\) has no absolutely irreducible factors in \(k[X,Y]\) except for scalar multiples of \(X-Y\). When \(k\) is finite, this condition is equivalent to saying that the map \(\alpha\mapsto f(\alpha)\) induces a bijection on an infinite algebraic extension of \(k\).
In this paper, the authors produce a new family of polynomials \(f(X)\) over fields of characteristic 2 which are exceptional and with degree not a power of 2; precisely \(\deg(f(X))= 2^{e-1}(2^e- 1)\), where \(e> 1\) is odd. They also prove that this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic.

MSC:

12F10 Separable extensions, Galois theory
12E05 Polynomials in general fields (irreducibility, etc.)