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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 \(\geq 4\).

11T06 Polynomials over finite fields
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