### Abstract

In this paper the following result is obtained. THEOREM. Let r be any positiveinteger; in all but finitely many finite fields k, of odd characteristic, for every polynomial f(x) ∈ k[x] of degree r that is not of the form ∝(g(x))^{2} or ∝x(g(x))^{2}, there exists a primitive root β ∈ k such that f(β) is a square in k. As a result of this and some computation we shallsee that for every finite field k of characteristic ≠ 2 or 3, there exists a primitive root β ∈ k such that — (∝^{2} + ∝ + 1) = β^{2} for some ek; also everylinear polynomial with nonzero constant term in the finite field k of odd characteristic represents both nonzero squares and nonsquares at primitive roots of k unless k = GF(3), GF(5) or GF(7).

Original language | English (US) |
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Pages (from-to) | 123-137 |

Number of pages | 15 |

Journal | Pacific Journal of Mathematics |

Volume | 98 |

Issue number | 1 |

Publication status | Published - 1982 |

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### ASJC Scopus subject areas

- Mathematics(all)