Congruence modulo a prime number
A congruence in which the modulus is a prime number. A distinguishing feature of the theory of congruences modulo a prime number is the fact that the residue classes modulo form a finite field of
elements. Congruences modulo a prime number can therefore be treated as equations over finite prime fields and algebraic-geometric methods, as well as methods of number theory, can be used to study them.
One of the basic questions in the theory of congruences with one variable , which is of great significance to algebraic number theory, coding theory and other branches of mathematics, is the question of the study of the laws of decomposition
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modulo a prime number , of arbitrary integer polynomials
into irreducible factors.
A second basic question in the theory of congruences modulo a prime number with
variables is the question of the number of solutions of a congruence equation
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when
vary independently of each other over either the whole set of residue classes modulo
(problems of complete residue systems), or over a particular part of it (problems of incomplete residue systems).
The first results of the research into the question of the number of solutions of quadratic and bi-quadratic congruences with two variables were obtained by C.F. Gauss [1] and J.L. Lagrange [2]. E. Artin [3] established a link between the problem of the number of solutions of the hyper-elliptic congruences (
) on a complete residue system modulo the prime number
and the Riemann hypothesis for
-functions of algebraic function fields with a finite field of constants, which were introduced by him. In particular, he stated the hypothesis that for the number
of solutions of the congruence
(
), where the polynomial
is not the square of another polynomial modulo
, the estimate
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is correct (here is the integer part of the number
).
Artin's hypothesis was first proved by H. Hasse [6] for the case of the elliptic congruences
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A. Weil [8] subsequently extended the method of Hasse to cover the general case and obtained the estimate
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for the number of solutions of the equation
in elements of the field
, consisting of
elements, where
is an absolutely-irreducible polynomial with coefficients from
. The Hasse–Weil method is complicated and requires the use of modern abstract algebraic geometry. A simple and purely arithmetic method of proving the results of Hasse and Weil can be found in [7].
Congruences modulo a prime number with variables are less widely studied. The following theorem can be used here as a general result. Let
be an absolutely-irreducible polynomial with integer coefficients. Then for the number
of solutions of the congruence
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the estimate
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holds, where the constant does not depend on
. A better estimate has been obtained by P. Deligne [9].
For results on congruences modulo a prime number on an incomplete residue system, see Vinogradov hypotheses; Two-term congruence; Distribution of power residues and non-residues.
References
[1] | C.F. Gauss, "Untersuchungen über höhere Arithmetik" , Springer (1889) (Translated from Latin) |
[2] | J.L. Lagrange, "Démonstration d'un théorème d'arithmétique" , Oeuvres , 3 , Paris (1869) pp. 189–201 |
[3] | E. Artin, "Quadratische Körper in Gebiete der höheren Kongruenzen II" Math. Z. , 19 (1924) pp. 207–246 |
[4] | I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian) |
[5] | H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) |
[6] | H. Hasse, "Abstrakte Begründung der komplexen multiplication und Riemannsche Vermutung in Funktionenkörpern" Abh. Math. Sem. Hamburg Univ. , 10 (1934) pp. 325–347 |
[7] | S.A. Stepanov, "A constructive method in the theory of equations over finite fields" Trudy Mat. Inst. Steklov. , 132 (1973) pp. 237–246 (In Russian) |
[8] | A. Weil, "Courbes algébriques et variétés abéliennes. Sur les courbes algébriques et les varietés qui s'en deduisent" , Hermann (1948) |
[9] | P. Deligne, "La conjecture de Weil I" Publ. Math. IHES , 43 (1974) pp. 273–307 |
Comments
A polynomial over
is absolutely irreducible if it is still irreducible over the algebraic closure
of
. For some material on polynomials over finite fields and their factorizations in the context of coding theory cf. [a1].
References
[a1] | F.J. MacWilliams, N.J.A. Sloane, "The theory of error-correcting codes" , I-II , North-Holland (1977) |
[a2] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 |
Congruence modulo a prime number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence_modulo_a_prime_number&oldid=18000