Abstract
A classical conjecture of E. Artin[Ar] predicts that any integer a ≠ ±1 or a perfect square is a primitive root (mod p) for infinitely many primes p. This conjecture is still open. In 1967, Hooley[H] proved the conjecture assuming the (as yet) unresolved generalized Riemann hypothesis for Dedekind zeta functions of certain number fields.
The first author, an undergraduate, would like to thank the second author for giving him the opportunity of being a part of this research project. Research of the second author was partially supported by NSERC.
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Reference
E. Artin, Collected Papers. Addison-Wesley, 1965.
H. Bilharz, Primdivisoren mit vorgeberger Primitivwurzel. Math. Annalen 114 (1937), 476–492.
E. Bombieri, Friedlander, J. and Iwaniec, H., Primes in arithmetic progressions to large moduli. Acta Math. 370 (1986), 203–251.
H. Davenport, On Primitive Roots in Finite Fields. Quart. J. Math. (2) 8 (1937), 308–312. (See also Collected Papers,vol. 4, p. 1557–1561.)
R. Gupta, Ram Murty, M. and Kumar Murty, V., The Euclidean algorithm for S-integers. In Number Theory, 189–201 (H. Kisilevsky and J. Labute, eds.). (Canadian Mathematical Society Conference Proceedings 7) (1987).
C. Hooley, On Artin’s conjecture. J. Reine. Angew. Math. 225 (1967), 209–220.
D.R. Heath-Brown, Artin’s conjecture for primitive roots. Quart. J. Math. Oxford (2) 37 (1986), 27–38.
G.H. Hardy and Wright, E.M., An Introduction to the Theory of Numbers, 5th ed. Oxford University Press, 1979.
K. Ireland and Rosen, M., A Classical Introduction to Modern Number Theory. SpringerVerlag, 1982.
D. Lorenzini, An Invitation to Arithmetic Geometry, vol. 9. Graduate Studies in Mathematics, American Math Society, 1996, pp. 354–360.
M. Ram Murty, Artin’s conjecture and elliptic analogues. In Sieve Methods, Exponential Sums, and their Applications in Number Theory, 325–344. ( G.R.H. Greaves, G. Harman, and M.N. Huxley, eds.). Cambridge University Press, 1996.
M. Ram Murty, Artin’s conjecture for primitive roots. Math. Intelligencer 10 (1988), 59–67.
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© 2000 Hindustan Book Agency (India) and Indian National Science Academy
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Jensen, E., Murty, M.R. (2000). Artin’s Conjecture for Polynomials Over Finite Fields. In: Bambah, R.P., Dumir, V.C., Hans-Gill, R.J. (eds) Number Theory. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7023-8_10
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DOI: https://doi.org/10.1007/978-3-0348-7023-8_10
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