Numbers of solutions of equations in finite fields

  title={Numbers of solutions of equations in finite fields},
  author={Andr{\'e} Weil},
  journal={Bulletin of the American Mathematical Society},
  • 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… 

The number of solutions of certain diagonal equations over finite fields

On the asymptotic distribution of the elementary symmetric functions \pmod

in finite fields. Niven and the author [4] solved the problem for the equation A„=a (mod m), where An is a determinant of order n in the independent variables xa (i, j = 1, ■ • ■ , n). In the cases

Solutions to Systems of Equations over Finite Fields

OF DISSERTATION SOLUTIONS TO SYSTEMS OF EQUATIONS OVER FINITE FIELDS This dissertation investigates the existence of solutions to equations over finite fields with an emphasis on diagonal equations.

Congruences in algebraic number fields involving sums of similar powers

where t is uniquely determined, and t is uniquely determined (mod P) if tOX. We discuss briefly the main results of the paper. An exact formula for Q,(p) involving the generalized Jacobi sum (2.9) is

On the Number and Distribution of Simultaneous Solutions to Diagonal Congruences

Two aspects, and their connections, of the problem of enumerating solutions to certain systems of congruences are explored in this paper. Although slightly more general cases are mentioned, the basic

Sieve methods for varieties over finite fields and arithmetic schemes

Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field

The Weil Conjectures

  • B. Hayes
  • Mathematics
    Notices of the American Mathematical Society
  • 2021
The Weil conjectures constitute one of the central landmarks of 20th century algebraic geometry: not only was their proof a dramatic triumph, but they served as a driving force behind a striking

Algebraic cycles and values of L-functions.

Let X be a smooth projective algebraic variety of dimension d over a number field k, and let n ̂ 0 be an integer, /-adic cohomology in degree /i, H(X^, Ot), is a representation space for Gal (Jc/k)

Number of Points on the Projective Curves ... Defined over Finite Fields, l an Odd Prime

The number of points on the curve aY e=bXe+c (abc{0) defined over a finite field Fq , q#1 (mod e), is known to be obtainable in terms of Jacobi sums and cyclotomic numbers of order e with respect to


Given an abelian variety A over a function field K = k(C) with C an absolutely irreducible, smooth, proper curve over a field k, it is natural to ask about the behavior of the Mordell-Weil group of A



On the Existence of Solutions of Certain Equations in a Finite Field.

  • L. HuaH. S. Vandiver
  • Biology
    Proceedings of the National Academy of Sciences of the United States of America
  • 1948
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

  • Proc. Nat. Acad. Sci. U.S.A. vol
  • 1949

J. Math. Pures Appl. J. Math. Pures Appl. vol

  • J. Math. Pures Appl. J. Math. Pures Appl. vol
  • 1837

Added in proof. Results, substantially identical to our formula (3), have just been published by

  • Proc. Nat. Acad. Sci. U.S.A. vol
  • 1949

J. Reine Angew. Math. vol

  • J. Reine Angew. Math. vol
  • 1922