TY - JOUR

T1 - Polynomials that represent quadratic residues at primitive roots

AU - Madden, Daniel J.

AU - VÉlez, William Yslas

PY - 1982/1

Y1 - 1982/1

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).

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U2 - 10.2140/pjm.1982.98.123

DO - 10.2140/pjm.1982.98.123

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 -