# zbMATH — the first resource for mathematics

When does a polynomial over a finite field permute the elements of the field? II. (English) Zbl 0777.11054
A polynomial over a finite field $$F_ q$$, $$q$$ a power of a prime, is a permutation polynomial (PP) if it induces a 1-1 mapping on $$F_ q$$. A brief survey of the main known classes of PP’s is given, following on from the earlier survey [R. Lidl and G. L. Mullen, Am. Math. Mon. 95, 243-246 (1988; Zbl 0653.12010)]. Progress on two of those problems has been particularly significant and is noted (using the notation of the earlier survey):
P8. Chowla and Zassenhaus conjecture: If $$p$$ is a sufficiently large prime and $$f(x)$$ of degree $$\geq 2$$ permutes $$F_ p$$, then $$f(x)+ax$$ with $$0<a<p$$ is not a PP of $$F_ p$$.
P9. Carlitz conjecture: For each positive integer $$k$$, there is a constant $$C_ k$$ such that for each finite field of odd order $$q>C_ k$$, there does not exist a PP of degree $$k$$ over $$F_ q$$.

##### MSC:
 11T06 Polynomials over finite fields
Full Text: