### 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) |
---|---|

Pages (from-to) | 123-137 |

Number of pages | 15 |

Journal | Pacific Journal of Mathematics |

Volume | 98 |

Issue number | 1 |

State | Published - 1982 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Polynomials that represent quadratic residues at primitive roots.** / Madden, Daniel; Velez, William Yslas.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 98, no. 1, pp. 123-137.

}

TY - JOUR

T1 - Polynomials that represent quadratic residues at primitive roots

AU - Madden, Daniel

AU - Velez, William Yslas

PY - 1982

Y1 - 1982

N2 - 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) = β2 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).

AB - 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) = β2 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).

UR - http://www.scopus.com/inward/record.url?scp=84972540037&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972540037&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84972540037

VL - 98

SP - 123

EP - 137

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -