On the solutions of the equation in a finite field

Authors:
Wen Fong Ke and Hubert Kiechle

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1331-1339

MSC:
Primary 11D79; Secondary 11T30

DOI:
https://doi.org/10.1090/S0002-9939-1995-1234628-4

MathSciNet review:
1234628

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Abstract | References | Similar Articles | Additional Information

Abstract: An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on *m* and the characteristic of the field, holds.

**[1]**J. R. Clay,*Circular block designs from planar near rings*, Ann. Discrete Math.**37**(1988), 95-106. MR**931309 (89g:05019)****[2]**-,*Nearrings*:*geneses and applications*, Oxford Univ. Press, Oxford, 1992. MR**1206901 (94b:16001)****[3]**J. R. Clay and H. Kiechle,*Linear codes from planar near rings and Möbius planes*, Algebras Groups Geom.**10**(1993), 333-344. MR**1250686 (94k:51006)****[4]**L. E. Dickson,*Cyclotomy, higher congruences, and Waring's problem*, Amer. J. Math.**57**(1935), 391-424. MR**1507083****[5]**-,*Congruences involving only e-th powers*, Acta Arith.**1**(1936), 161-167.**[6]**O. B. Faircloth,*On the number of solutions of some general types of equations in a finite field*, Canad. J. Math.**4**(1952), 343-351. MR**0047694 (13:915d)****[7]**L.-K. Hua and H. S. Vandiver,*Characters over certain types of rings with applications to the theory of equations in a finite field*, Proc. Nat. Acad. Sci. U.S.A.**35**(1949), 94-99. MR**0028895 (10:515c)****[8]**-,*On the number of solutions of some trinomial equations in a finite field*, Proc. Nat. Acad. Sci. U.S.A.**35**(1949), 477-481. MR**0032679 (11:329g)****[9]**K. F. Ireland and M. I. Rosen,*A classical introduction to modern number theory*, second ed., Springer-Verlag, Berlin, Heidelberg, and New York, 1990. MR**1070716 (92e:11001)****[10]**J.-R. Joly,*Équations et variétés algébriques sur un corps fini*, Enseign. Math. (2)**19**(1973), 1-117.**[11]**W.-F. Ke,*Structures of circular planar nearings*, Ph.D. dissertaton, Univ. Arizona, Tucson, 1992.**[12]**W.-F. Ke and H. Kiechle,*Combinatorial properties of ring generated circular planar nearings*(submitted).**[13]**D. H. Lehmer,*The number of solutions of a certain congruence involving the sum of like powers*, Utilitas Math.**39**(1991), 65-89. MR**1119764 (92f:11005)****[14]**H. H. Mitchell,*On the congruence**in a Galois field*, Ann. of Math. (2)**18**(1917), 120-131. MR**1503593****[15]**M. C. Modisett,*A characterization of the circularity of certain balanced incomplete block designs*, Ph.D. dissertation, Univ. Arizona, Tucson, 1988.**[16]**-,*A characterizatin of the circularity of balanced incomplete block designs*, Utilitas Math.**35**(1989), 83-94. MR**992392 (90c:05028)****[17]**Q. Sun,*The diagonal equation over a finite field*, Sichuan Daxue Xuebao**26**(1989), 159-162. (Chinese) MR**1017018 (90i:11145)****[18]**A. Weil,*Number of solutions of equations in a finite field*, Bull. Amer. Math. Soc.**55**(1949), 497-508. MR**0029393 (10:592e)****[19]**J. Wolfmann,*The number of solutions of certain diagonal equations over finite fields*, J. Number Theory**42**(1992), 247-257. MR**1189504 (94j:11055)**

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1234628-4

Article copyright:
© Copyright 1995
American Mathematical Society