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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249001.png" /> form a finite field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249002.png" /> 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.
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One of the basic questions in the theory of congruences with one variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249003.png" />, which is of great significance to [[Algebraic number theory|algebraic number theory]], coding theory and other branches of mathematics, is the question of the study of the laws of decomposition
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249004.png" /></td> </tr></table>
+
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  $  p $
 +
form a finite field of  $  p $
 +
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.
  
modulo a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249005.png" />, of arbitrary integer polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249006.png" /> into irreducible factors.
+
One of the basic questions in the theory of congruences with one variable  $  x $,
 +
which is of great significance to [[Algebraic number theory|algebraic number theory]], coding theory and other branches of mathematics, is the question of the study of the laws of decomposition
  
A second basic question in the theory of congruences modulo a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249007.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249008.png" /> variables is the question of the number of solutions of a [[Congruence equation|congruence equation]]
+
$$
 +
f( x)  \equiv  f _ {1} ( x) \dots f _ {r} ( x)  (  \mathop{\rm mod}  p) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c0249009.png" /></td> </tr></table>
+
modulo a prime number  $  p $,
 +
of arbitrary integer polynomials  $  f( x) $
 +
into irreducible factors.
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490010.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490011.png" /> vary independently of each other over either the whole set of residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490012.png" /> (problems of complete residue systems), or over a particular part of it (problems of incomplete residue systems).
+
A second basic question in the theory of congruences modulo a prime number  $  p $
 +
with  $  n \geq  2 $
 +
variables is the question of the number of solutions of a [[Congruence equation|congruence equation]]
  
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 [[#References|[1]]] and J.L. Lagrange [[#References|[2]]]. E. Artin [[#References|[3]]] established a link between the problem of the number of solutions of the hyper-elliptic congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490014.png" />) on a complete residue system modulo the prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490015.png" /> and the Riemann hypothesis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490016.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490017.png" /> of solutions of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490018.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490019.png" />), where the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490020.png" /> is not the square of another polynomial modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490021.png" />, the estimate
+
$$
 +
f( x _ {1} \dots x _ {n} ) \equiv  0 ( \mathop{\rm mod}  p),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490022.png" /></td> </tr></table>
+
when  $  x _ {i} $
 +
$  ( 1 \leq  i \leq  n) $
 +
vary independently of each other over either the whole set of residue classes modulo  $  p $(
 +
problems of complete residue systems), or over a particular part of it (problems of incomplete residue systems).
  
is correct (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490023.png" /> is the integer part of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490024.png" />).
+
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 [[#References|[1]]] and J.L. Lagrange [[#References|[2]]]. E. Artin [[#References|[3]]] established a link between the problem of the number of solutions of the hyper-elliptic congruences  $  y  ^ {2} \equiv f( x) $(
 +
$  \mathop{\rm mod}  p $)
 +
on a complete residue system modulo the prime number  $  p $
 +
and the Riemann hypothesis for  $  \zeta $-
 +
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  $  N _ {p} $
 +
of solutions of the congruence  $  y  ^ {2} \equiv f( x) $(
 +
$  \mathop{\rm mod}  p $),
 +
where the polynomial  $  f( x) = x  ^ {n} + a _ {1} x  ^ {n-} 1 + \dots + a _ {n} $
 +
is not the square of another polynomial modulo  $  p $,
 +
the estimate
 +
 
 +
$$
 +
| N _ {p} - p |  \leq  2 \left [ n-
 +
\frac{1}{2}
 +
\right ] p  ^ {1/2}
 +
$$
 +
 
 +
is correct (here $  [ x] $
 +
is the integer part of the number $  x $).
  
 
Artin's hypothesis was first proved by H. Hasse [[#References|[6]]] for the case of the elliptic congruences
 
Artin's hypothesis was first proved by H. Hasse [[#References|[6]]] for the case of the elliptic congruences
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490025.png" /></td> </tr></table>
+
$$
 +
y  ^ {2}  \equiv  x  ^ {3} + ax + b  (  \mathop{\rm mod}  p).
 +
$$
  
 
A. Weil [[#References|[8]]] subsequently extended the method of Hasse to cover the general case and obtained the estimate
 
A. Weil [[#References|[8]]] subsequently extended the method of Hasse to cover the general case and obtained the estimate
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490026.png" /></td> </tr></table>
+
$$
 +
| N _ {q} - q |  \leq  c( f  ) q  ^ {1/2}
 +
$$
  
for the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490027.png" /> of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490028.png" /> in elements of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490029.png" />, consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490030.png" /> elements, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490031.png" /> is an absolutely-irreducible polynomial with coefficients from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490032.png" />. 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 [[#References|[7]]].
+
for the number $  N _ {q} $
 +
of solutions of the equation $  f( x, y) = 0 $
 +
in elements of the field $  F _ {q} $,  
 +
consisting of $  q = p  ^ {r} $
 +
elements, where $  f( x, y) $
 +
is an absolutely-irreducible polynomial with coefficients from $  F _ {q} $.  
 +
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 [[#References|[7]]].
  
Congruences modulo a prime number with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490033.png" /> variables are less widely studied. The following theorem can be used here as a general result. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490034.png" /> be an absolutely-irreducible polynomial with integer coefficients. Then for the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490035.png" /> of solutions of the congruence
+
Congruences modulo a prime number with $  n $
 +
variables are less widely studied. The following theorem can be used here as a general result. Let $  f( x _ {1} \dots x _ {n} ) $
 +
be an absolutely-irreducible polynomial with integer coefficients. Then for the number $  N _ {p} $
 +
of solutions of the congruence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490036.png" /></td> </tr></table>
+
$$
 +
f( x _ {1} \dots x _ {n} )  \equiv  0 (  \mathop{\rm mod}  p),\ \
 +
n \geq  2,
 +
$$
  
 
the estimate
 
the estimate
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490037.png" /></td> </tr></table>
+
$$
 +
| N _ {p} - p  ^ {n-} 1 |  \leq  c( f  ) p  ^ {n-} 1- 1/2
 +
$$
  
holds, where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490038.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490039.png" />. A better estimate has been obtained by P. Deligne [[#References|[9]]].
+
holds, where the constant c( f  ) $
 +
does not depend on $  p $.  
 +
A better estimate has been obtained by P. Deligne [[#References|[9]]].
  
 
For results on congruences modulo a prime number on an incomplete residue system, see [[Vinogradov hypotheses|Vinogradov hypotheses]]; [[Two-term congruence|Two-term congruence]]; [[Distribution of power residues and non-residues|Distribution of power residues and non-residues]].
 
For results on congruences modulo a prime number on an incomplete residue system, see [[Vinogradov hypotheses|Vinogradov hypotheses]]; [[Two-term congruence|Two-term congruence]]; [[Distribution of power residues and non-residues|Distribution of power residues and non-residues]].
Line 43: Line 103:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C.F. Gauss, "Untersuchungen über höhere Arithmetik" , Springer (1889) (Translated from Latin) {{MR|0188045}} {{ZBL|21.0166.04}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.L. Lagrange, "Démonstration d'un théorème d'arithmétique" , ''Oeuvres'' , '''3''' , Paris (1869) pp. 189–201</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Artin, "Quadratische Körper in Gebiete der höheren Kongruenzen II" ''Math. Z.'' , '''19''' (1924) pp. 207–246</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian) {{MR|0807530}} {{ZBL|0577.01049}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) {{MR|0153659}} {{ZBL|1038.11500}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Hasse, "Abstrakte Begründung der komplexen multiplication und Riemannsche Vermutung in Funktionenkörpern" ''Abh. Math. Sem. Hamburg Univ.'' , '''10''' (1934) pp. 325–347</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A. Weil, "Courbes algébriques et variétés abéliennes. Sur les courbes algébriques et les varietés qui s'en deduisent" , Hermann (1948)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Deligne, "La conjecture de Weil I" ''Publ. Math. IHES'' , '''43''' (1974) pp. 273–307 {{MR|0340258}} {{ZBL|0314.14007}} {{ZBL|0287.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C.F. Gauss, "Untersuchungen über höhere Arithmetik" , Springer (1889) (Translated from Latin) {{MR|0188045}} {{ZBL|21.0166.04}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.L. Lagrange, "Démonstration d'un théorème d'arithmétique" , ''Oeuvres'' , '''3''' , Paris (1869) pp. 189–201</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Artin, "Quadratische Körper in Gebiete der höheren Kongruenzen II" ''Math. Z.'' , '''19''' (1924) pp. 207–246</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian) {{MR|0807530}} {{ZBL|0577.01049}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) {{MR|0153659}} {{ZBL|1038.11500}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Hasse, "Abstrakte Begründung der komplexen multiplication und Riemannsche Vermutung in Funktionenkörpern" ''Abh. Math. Sem. Hamburg Univ.'' , '''10''' (1934) pp. 325–347</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A. Weil, "Courbes algébriques et variétés abéliennes. Sur les courbes algébriques et les varietés qui s'en deduisent" , Hermann (1948)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Deligne, "La conjecture de Weil I" ''Publ. Math. IHES'' , '''43''' (1974) pp. 273–307 {{MR|0340258}} {{ZBL|0314.14007}} {{ZBL|0287.14001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490040.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490041.png" /> is absolutely irreducible if it is still irreducible over the algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024900/c02490043.png" />. For some material on polynomials over finite fields and their factorizations in the context of coding theory cf. [[#References|[a1]]].
+
A polynomial $  f ( x , y ) $
 +
over $  F _ {q} $
 +
is absolutely irreducible if it is still irreducible over the algebraic closure $  \overline{F}\; _ {q} $
 +
of $  F _ {q} $.  
 +
For some material on polynomials over finite fields and their factorizations in the context of coding theory cf. [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.J. MacWilliams, N.J.A. Sloane, "The theory of error-correcting codes" , '''I-II''' , North-Holland (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.J. MacWilliams, N.J.A. Sloane, "The theory of error-correcting codes" , '''I-II''' , North-Holland (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}} </TD></TR></table>

Latest revision as of 17:46, 4 June 2020


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 $ p $ form a finite field of $ p $ 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 $ x $, 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

$$ f( x) \equiv f _ {1} ( x) \dots f _ {r} ( x) ( \mathop{\rm mod} p) , $$

modulo a prime number $ p $, of arbitrary integer polynomials $ f( x) $ into irreducible factors.

A second basic question in the theory of congruences modulo a prime number $ p $ with $ n \geq 2 $ variables is the question of the number of solutions of a congruence equation

$$ f( x _ {1} \dots x _ {n} ) \equiv 0 ( \mathop{\rm mod} p), $$

when $ x _ {i} $ $ ( 1 \leq i \leq n) $ vary independently of each other over either the whole set of residue classes modulo $ p $( 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 $ y ^ {2} \equiv f( x) $( $ \mathop{\rm mod} p $) on a complete residue system modulo the prime number $ p $ and the Riemann hypothesis for $ \zeta $- 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 $ N _ {p} $ of solutions of the congruence $ y ^ {2} \equiv f( x) $( $ \mathop{\rm mod} p $), where the polynomial $ f( x) = x ^ {n} + a _ {1} x ^ {n-} 1 + \dots + a _ {n} $ is not the square of another polynomial modulo $ p $, the estimate

$$ | N _ {p} - p | \leq 2 \left [ n- \frac{1}{2} \right ] p ^ {1/2} $$

is correct (here $ [ x] $ is the integer part of the number $ x $).

Artin's hypothesis was first proved by H. Hasse [6] for the case of the elliptic congruences

$$ y ^ {2} \equiv x ^ {3} + ax + b ( \mathop{\rm mod} p). $$

A. Weil [8] subsequently extended the method of Hasse to cover the general case and obtained the estimate

$$ | N _ {q} - q | \leq c( f ) q ^ {1/2} $$

for the number $ N _ {q} $ of solutions of the equation $ f( x, y) = 0 $ in elements of the field $ F _ {q} $, consisting of $ q = p ^ {r} $ elements, where $ f( x, y) $ is an absolutely-irreducible polynomial with coefficients from $ F _ {q} $. 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 $ n $ variables are less widely studied. The following theorem can be used here as a general result. Let $ f( x _ {1} \dots x _ {n} ) $ be an absolutely-irreducible polynomial with integer coefficients. Then for the number $ N _ {p} $ of solutions of the congruence

$$ f( x _ {1} \dots x _ {n} ) \equiv 0 ( \mathop{\rm mod} p),\ \ n \geq 2, $$

the estimate

$$ | N _ {p} - p ^ {n-} 1 | \leq c( f ) p ^ {n-} 1- 1/2 $$

holds, where the constant $ c( f ) $ does not depend on $ p $. 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) MR0188045 Zbl 21.0166.04
[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) MR0807530 Zbl 0577.01049
[5] H. Hasse, "Zahlentheorie" , Akademie Verlag (1963) MR0153659 Zbl 1038.11500
[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 MR0340258 Zbl 0314.14007 Zbl 0287.14001

Comments

A polynomial $ f ( x , y ) $ over $ F _ {q} $ is absolutely irreducible if it is still irreducible over the algebraic closure $ \overline{F}\; _ {q} $ of $ F _ {q} $. 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 MR0568909 Zbl 0423.10001
How to Cite This Entry:
Congruence modulo a prime number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence_modulo_a_prime_number&oldid=23797
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article