Fields that admit a nonlinear permutation polynomial (Q1119970)

A permutation polynomial f on a commutative unitary ring R is a polynomial for which the induced function \(\alpha_ f\) on R \((\alpha_ f(r)=f(r)\) for all \(r\in R)\) is a permutation. Denoting the (cancellative) monoid of these polynomials by P(R) and that of the linear permutation polynomials \(aX+b\), a being a unit of R, by L(R), conditions are derived under which \(L(R)<P(R)\) holds, in particular for R being a field. - Classes of fields fulfilling this relation are the one of real closed fields (theorem 9) and that of perfect fields of characteristic \(p\neq 0.\)- On the other hand the algebraically closed fields of characteristic 0 (proposition 6) and the quotient fields of valuation domains with principal maximal ideal (theorem 2) admit no nonlinear permutation polynomial. Furthermore a method is derived to find to given \(f\in P(F)\) subfields of F admitting f as permutation polynomial, too, thus producing new classes of fields in the first case.