Smallest polynomial nonresidue and incomplete Gaussian sums in finite fields (Q1326008)

The paper deals with the concept of quadratic residues and nonresidues in the ring \(\mathbb{F}_ q [x]\) of polynomials over a finite field \(\mathbb{F}_ q\) modulo an irreducible polynomial \(P\). Let \(s(q,P)\) be the smallest integer, such that among the unitary polynomials of degree \(s\), there is a quadratic nonresidue modulo \(P\). The author proves the following theorem: For any \(0 < \varepsilon < 1\) and almost all (except for a finite number) fields \(\mathbb{F}_ q\) there exists a number \(n_ 0 = n_ 0 (\varepsilon,q)\) such that \(\max s(q,P) < \varepsilon n\) for all \(n>n_ 0\). Here, the maximum is taken over all polynomials \(P\) of degree \(n\) irreducible over \(\mathbb{F}_ q\). The generalizations and modifications of the sums used in the proof of this theorem lead to incomplete Gaussian sums.