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On a problem of L. Carlitz. (Sur un problème de L. Carlitz.) (French) Zbl 0812.11068
In [Duke Math. J. 19, 471-474 (1952; Zbl 0049.032)] L. Carlitz showed that, for any integer $$d\geq 2$$, there exists an integer $$N(d)$$ such that, for any finite field $$\mathbb{F}_ q$$ with $$q$$ odd and $$q>N(d)$$, any $$f$$ of $$\mathbb{F}_ q [X]$$ such that for every $$x$$ of $$\mathbb{F}_ q$$, $$f(x)$$ is a square of $$\mathbb{F}_ q$$, is a square in $$\mathbb{F}_ q [X]$$ and altogether, L. Carlitz solved an old conjecture of Dickson, see also his paper [Duke Math. J. 14, 1139-1140 (1947; Zbl 0031.10505)].
This problem of Dickson and Carlitz is considered again and it is shown that for $$d$$ odd, $$N(d)$$ can be given the value $$d^ 2$$ and for $$d$$ even and $$\geq 4$$, it can be given the value $$(d-1)^ 2$$ and that these values of $$N(d)$$ are generally best possible. We also show, using an elementary method analogous to one in a 1973 Stark paper, that, when the problem is restricted to prime fields, one can choose $$N(d)$$ to be $$(d^ 2+ 2d-1) /2$$ for odd $$d$$’s and $$(d^ 2- d-4)/2$$ when $$d$$ is even and $$\geq 4$$.

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