Numbers of solutions of equations in finite fields

@article{Weil1949NumbersOS,
  title={Numbers of solutions of equations in finite fields},
  author={Andr{\'e} Weil},
  journal={Bulletin of the American Mathematical Society},
  year={1949},
  volume={55},
  pages={497-508}
}
  • A. Weil
  • Published 1 May 1949
  • Mathematics
  • Bulletin of the American Mathematical Society
Such equations have an interesting history. In art. 358 of the Disquisitiones [1, a], Gauss determines the Gaussian sums (the so-called cyclotomic “periods”) of order 3, for a prime of the form p = 3n + 1, and at the same time obtains the number of solutions for all congruences ax− by ≡ 1 (mod p). He draws attention himself to the elegance of his method, as well as to its wide scope; it is only much later, however, viz. in his memoir on biquadratic residues [1, b], that he gave in print another… 
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References

SHOWING 1-10 OF 13 REFERENCES
On the Existence of Solutions of Certain Equations in a Finite Field.
  • L. Hua, H. S. Vandiver
  • Biology
    Proceedings of the National Academy of Sciences of the United States of America
  • 1948
TLDR
This paper aims to provide a history of anthocyanin pigments of plants and their role in the biochemistry of fruit and vegetable establishment and its role in disease.
Some problems of ‘Partitio Numerorum’: IV. The singular series in Waring’s Problem and the value of the number G (k)
In this memoir we continue the investigations initiated in two earlier memoirs bearing a similar title, and complete the proof of all the assertions which they contain 1). We shall assume throughout
Proc. Nat. Acad. Sci. U.S.A. vol
  • Proc. Nat. Acad. Sci. U.S.A. vol
  • 1948
THE UNIVERSITY OF CHICAGO 5 Added in proof. Results, substantially identical to our formula (3), have just been published by
  • Proc. Nat. Acad. Sci. U.S.A
  • 1890
J. Math. Pures Appl. J. Math. Pures Appl. vol
  • J. Math. Pures Appl. J. Math. Pures Appl. vol
  • 1837
a) J
  • Math. Pures Appl. vol. 2 (1837) pp. 253-292; (b) J. Math. Pures Appl. vol. 3
  • 1838
THE UNIVERSITY OF CHICAGO 5 Added in proof. Results, substantially identical to our formula
  • Proc. Nat. Acad. Sci. U.S.A. vol
  • 1949
Added in proof. Results, substantially identical to our formula (3), have just been published by
  • Proc. Nat. Acad. Sci. U.S.A. vol
  • 1949
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